Saxon Math 8/7 (Student Edition) [Third edition] 1565775090, 9781565775091

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Student Edition

MATH

THIS BOOK

IS

THE PROPERT Y OF:

STATE

BookNo^MSi

PROVINCE

Enter information

COUNTY

in

PARISH

**'

spaces




365/7 = 365

Use words

Then

(b)

each missing number?

52

•

7

=

7m

to describe the rule of the following sequence.

find the next three terms of the sequence.

...,10,8,6,4,2,... 29.

Name

(1.4)

,

30. (1 ' 4)

.

each

set of

,

numbers

illustrated:

.

(a)

{1,2,3,4,

(b)

{0, 1, 2, 3, ...|

(c)

{...,-2,-1, 0,

...}

1, 2, ...}

Use braces, an ellipsis, and negative even numbers.

digits to illustrate the set of

33

Lesson 6

LESSON

H|

Factors

Divisibility

•

WARM-UP Facts Practice: 30 Equations (Test B)

Mental Math: a.

$5.00 + $2.50

b.

$1.50 x 10

d.

450 + 35

e.

675 - 50

g.

9 X

5,

-

1,

+

4,

-r

1,

4,

-r

c. f.

X 5,

+

1,

-r

$1.00

750

-

-f

-

$0.45

10

4+

Problem Solving: If

there are twelve glubs in a lorn

how many glubs

and four lorns in a

dort,

then

are in half a dort?

NEW CONCEPTS Factors

Recall that factors are the

numbers multiplied

form a

to

product.

= 15

3

x 5

1

x 15

both 3 and

= 15

1

and 15

are factors of 15

numbers

1, 3, 5,

and 15 can serve

both

Therefore, each of the factor of 15.

5 are factors of 15

as a

Notice that 15 can be divided by 1, 3, 5, or 15 without a remainder. This leads us to another definition of factor.

The

number are the whole numbers number without a remainder.

factors of a

that divide the

For example, the numbers 1, 2, 5, and 10 are factors of 10 because each divides 10 without a remainder (that is, with a remainder of zero). 10 lJTo 10

_5 2JT0 10

2

1

5ll0 10

lOjlO 10

~0 +

As

a shorthand,

sequentially from

we

will use

left to right.

commas This

is

to separate operations to

be performed

not standard mathematical notation.

Saxon Math 8/7

Example

1

Solution

Example 2 Solution

List the

whole numbers

that are factors of 12.

The factors of 12 are the whole numbers that divide 12 with no remainder. They are 1, 2, 3, 4, 6, and 12. List the factors of 51.

As we

try to think of

whole numbers

no and

that divide 51 with

we may

think that 51 has only two factors, 1 51. However, there are actually four factors of 51. Notice that 3 and 17 are also factors of 51.

remainder,

17 3

is

a factor

of 51

—

common of 12

and and

1

factors, 1

and 51

2 3.

17

is

a factor

of 51

we

1, 3, 17,

and

51.

and 51 have two The greatest common factor (GCF)

because

is 3,

—

3)51

Thus, the four factors of 51 are

From examples

-*

see that 12

it is

common

the largest

factor of

both numbers.

Example 3 Solution

Find the greatest

We are asked to Here

30.

common

we

common

factor of 18

and

30.

find the largest factor (divisor) of both 18 and

list

the factors of both numbers, circling the

factors.

®,©,©,®, 9,18 of 30: ®,©,@, 5,®, 10,

Factors of 18: Factors

The Divisibility

greatest

common

factor of 18

and 30

15, 30

is 6.

As we saw in example 2, the number 51 can be divided by 1, 3, 17, and 51 with a remainder of zero. The capability of a whole number to be divided by another whole number with no remainder is called divisibility. Thus, 51 is divisible by 1, 3, 17, and 51. There are several methods for testing the divisibility of a number without actually performing the division. Listed below are methods for testing whether a number is divisible

by

2, 3, 4, 5, 6, 8, 9,

or 10.

Lesson 6

35

Tests for Divisibility

A number is

by

divisible

...

even.

2

if

the last digit

4

if

8

if

the last two digits can be divided by 4. the last three digits can be divided by 8.

5

if

the last digit

is

10

if

the last digit

is 0.

3

if

the

sum

6

if

the

number can be divided by

9

if

the

sum

is

or

5.

of the digits can be divided by of the digits

A number ending in

and by 3. can be divided by 9.

Solution

Which whole numbers from In the sense

used in

this

2

...

one zero is divisible by 2. two zeros is divisible by 2 and 4. three zeros is divisible by 2, 4, and

Example 4

3.

1 to

8.

10 are divisors of 9060?

problem, a divisor

is

a factor.

The

whole number. As we apply the tests for divisibility, we find that 9060 passes the tests for 2, and 10, but not for 8. The sum of its digits 5, 4, + 6 + 0) is 15, which can be divided by 3 but not by (9 + 9. Since 9060 is divisible by both 2 and 3, it is also divisible by 6. The only whole number from 1 to 10 we have not tried is 7, for which we have no simple test. We divide 9060 by 7 to find that 7 is not a divisor. We find that the numbers from 1 to 10 that are divisors of 9060 are 1, 2, 3, 4, 5, 6, and 10.

number

1 is

a divisor of any

LESSON PRACTICE Practice set*

List the a.

whole numbers

25

List the

b.

that are factors of each

24

c.

whole numbers from

1 to

number:

23

10 that are factors of each

number: d.

1260

g.

List the single-digit divisors of 1356.

h.

e.

The number 7000 numbers?

73,500

is

divisible

f.

3600

by which

single-digit

36

Saxon Math 8/7

common

1.

List all the

j.

Find the greatest

factors of 12

common

and

20.

factor (GCF) of 24

and

40.

MIXED PRACTICE Problem set

1. 111

2.

the product of 10 and 20 is divided by the and 30, what is the quotient? If

common

(a)

List all the

(b)

Find the greatest

factors of 30

and

sum

of 20

40.

(6)

3. (4)

4. t5)

Use braces, an ellipsis, and negative odd numbers. digits

List the

161

12,300.

141

factor of 30

and

40.

digits to illustrate the set of

write four hundred seven million, six thousand, nine hundred sixty-two.

Use

5.

6.

common

to

whole numbers from

1 to

10 that are divisors of

Replace the circle with the proper comparison symbol. Then write the comparison as a complete sentence using words to write the numbers.

-7Q-11 7. (6)

8.

The number 3456 numbers?

Show

divisible

is

this subtraction

by which

problem on a number

single-digit

line:

(4)

9.

Write 6400 in expanded notation.

(5)

Find each missing number: 10.

x + $4.60 = $10.00

(3)

= $50.00

(3)

(3)

15.

p - 3850 = 4500

(3)

12. 8z 14.

11. 13. (3)

1426 - k = 87

— 990

7

4 8

6

= 45

(3)

2 1

6 16. (3)

- = 8

32

8

9

+ n 60

2-5

37

17. ((a))

2.2.

=p 33

is.

:

ifni)

SI

19

:: r-o

_

21.

5-4

25

4c

c

7-11-13

-

24

.

r

-

:

;

-

z

"

-

:

Z

- S5 4

- 5 14

-

-

:

_

:

f4.

V

Name

:.'

10

the property illustrated by this equation, and

x

-

5

=

5a~

38

Saxon Math 8/7

LESSON Lines and Angles WARM-UP Facts Practice: 30 Equations (Test B)

Mental Math: a.

5

- 10

b.

$2.50 x 10

d.

340 + 25

e.

565 - 300

g.

Start

c. f.

$1.00 -

480

with the number of years in a decade, x

35(2

-f

10

7,

+

5,

3,

-1,-4. Problem Solving:

The sum of the counting numbers from 1 through is 10. What is the sum of the counting numbers

4

from

1

r5^

1+2

through 20?

+ 3 + 4

NEW CONCEPT We

world of three dimensions called space. We can measure the length, width, and depth of objects that occupy space. We can imagine a two-dimensional world called a plane, a flat world having length and width but not depth. Occupants of a two-dimensional world could not pass over or under other objects because, without depth, "over" and "under" would not exist. A one-dimensional world, a line, has length but neither width nor depth. Occupants of a onedimensional world could not pass over, under, or to either side of other objects. They could only move back and forth on live in a

their line.

we

study figures that have one dimension, two dimensions, and three dimensions, but we begin with a point, which has no dimensions. A point is an exact location in space and is unmeasurably small. We represent points with dots and usually name them with uppercase letters. Here we show point A: In geometry

A •

A

line contains

an

infinite

number

of points extending in

opposite directions without end. A line has one dimension, length. A line has no thickness. We can represent a line by sketching part of a line with two arrowheads. We identify a

— 39

Lesson 7

line

by naming two points on the

show

line

line in either order.

Here

we

AB (or line BA): B -•-

Line

The symbols

AB and BA

can be used to refer

A

ray

is

AB or

line

BA

(read "line

AB" and

"line

BA") also

to the line above.

a part of a line with one endpoint.

We

identify a ray

by naming the endpoint and then one other point on the Here we show ray AB [AB): A

B

•

•

ray.

—

Ray

AB

A

segment is a part of a line with two endpoints. We identify a segment by naming t he t wo endpoints in either order. Here we show segment AB [AB): A

B

•-

-•

Segment AB or segment BA

A segment has a specific length. We may refer to the length of segment AB by writing mAB, which means "the measure of segment AB," or by writing the letters AB without an overbar. Thus, both AB and mAB refer to the distance from point A to point B. We use this notation in the figure below to state that the sum of the lengths of the shorter segments equals the length of the longest segment.

C

B

A •-

AB

mAB Example

1

Solution

The

line is

AC

mBC

=

mAC

AB

(or

AB (or BA). The rays are AB

B

A

and BA The segment .

BA).

In the figure below, m

Solution

=

Use symbols to name a line, two rays, and a segment in the figure at right.

is

Example 2

+

BC

+

AB is

3

cm and AC is

7

cm. Find BC.

A

B

C

m

•

•

—

BC represents the length of segment BC. We are given that AB is 3 cm and AC is 7 cm. From the figure above, we see that AB + BC = AC. Therefore, we find that BC is 4 cm.

40

Saxon Math 8/7

A

plane is a flat surface that extends without end. It has two dimensions, length and width. A desktop occupies a part of a plane.

Two

once or do not cross at all. If two lines cross, we say that they intersect at one point. If two lines in a plane do not intersect, they remain the lines in the

same distance

same plane

apart

and

either cross

are called parallel lines.

Line

AB intersects

line

CD at

QR

is

point M.

R In this figure, line

parallel to line

SI

This statement can be written with symbols,

as

we show

here:

QR

II

ST

Lines that intersect and form "square corners" are perpendicular. The small square in the figure below indicates a "square corner."

M In this figure,

p

1

-•-

Q

line

line

MN

is

perpendicular to

PQ. This statement can be

symbols, as

we show

MN

written with

here:

1 PQ

Lines in a plane that are neither parallel nor perpendicular are oblique. In our figure showing intersecting lines, lines AB

and

CD are

An

angle

oblique.

is

formed by two rays that

common

endpoint. The angle right_is formed by the two rays

have

a

at

MD

and MB. The common endpoint Point

Ray

M

MD

is

M.

the vertex of the angle. are the sides of and ray is

MB

the angle. Angles may be named by listing the following points in order: a point on one ray, the vertex, and then a point on the other ray. So our angle

may angle

named BMD.

be

either angle

DMB

or

Angle

DMB BMD

or angle

41

Lesson 7

When

no chance of confusion, an angle may be named by only one point, the vertex. At right we have there

is

angle A.

Angle

A

Angle

1

An angle may also be named by placing a small letter or

number near

the vertex

and between the rays (in the interior of the angle). Here we see angle 1.

The symbol Z

is

often used instead of the

the three angles just

named could be

ZDMB ZA

word

angle. Thus,

referred to as:

read as "angle

DMB"

read as "angle

A"

read as "angle 1"

Zl

Angles are classified by their size. An angle formed by perpendicular rays is a right angle and is commonly marked with a small square at the vertex. An angle smaller than a right angle is an acute angle. An angle that forms a straight line is a straight angle. An angle smaller than a straight angle but larger than a right angle is an obtuse angle.

Acute

Right

Example

3

(a)

(b)

Which

line is parallel to line

Which line line AB?

is

Obtuse

Straight

AB?

perpendicular to

^—A

B c

c

c

Solution

(a)

(b)

Example 4

(or

DC)

is

parallel to line

(or

d5)

is

perpendicular to line AB.

There are several angles in (a)

(b) (c)

(d)

Solution

CD Line BD Line

(a)

(c)

D

AB.

this figure.

Name the straight angle. Name the obtuse angle. Name two right angles. Name two acute angles.

ZAMD (or ZDMA) 1. ZAMB (or ZBMA) 2. ZBMD (or ZDMB)

(b)

(d)

ZAMC (or ZCMA) 1. ZBMC (or ZCMB) 2. ZCMD (or ZDMC)

42

Saxon Math 8/7

On

we

with the force of gravity as vertical and objects aligned with the horizon as horizontal.

Example

5

earth

refer to objects aligned

A power

pole with two cross pieces can be represented by three segments. (a)

Name

a vertical segment.

(b)

Name

a horizontal segment.

(c)

Name

a

to

C

c

*-

-•

-+

D F

segment perpendicular

CD. B

Solution

(a)

(b) (c)

AB (or bX) CD (or DC) AB (or BA)

or

EF (or FE)

LESSON PRACTICE Practice set

a.

Name

a point on this figure that

is

not on ray BC: D

B

A

-•-

-•-

b. In this figure

XZ is

10 cm, and

X

YZ is

6 cm. Find

XY. Z

Y

c.

Draw two

d.

Draw two perpendicular

e.

Draw two lines that intersect but are not perpendicular. What word describes the relationship of these lines?

f.

Draw

parallel lines.

a right angle.

g.

Draw an

h.

Draw an obtuse

i.

Two

lines.

acute angle.

angle.

intersecting segments are

drawn on the board. One

segment is vertical and the other is horizontal. Are the segments parallel or perpendicular?

43

Lesson 7

MIXED PRACTICE Problem set

1. (3)

2.

If is

whole numbers the sum of the same two numbers?

the product of two one-digit

Name

is

35,

what

the property illustrated by this equation:

(2)

-5 3.

List the

whole number

•

= -5

1

divisors of 50.

(6)

4. t4)

5.

Use

digits

and symbols

"Two minus

to write

five equals

negative three."

Use only

digits

and commas

to write

90 million.

(5)

6. List

the single-digit factors of 924.

(6)

7.

Arrange these numbers in order from least to

greatest:

(4)

-10, 5,-7,

8. l2)

8,

0,-2

sequence. Then find the next three numbers in the sequence.

Use words

to describe the following

49, 64, 81, 100,

9. (7)

To build

...

Megan dug

holes in the ground to hold the posts upright. Then she nailed rails to connect the posts. Which fence parts were vertical, the posts or the a fence,

rails?

10. (a) List the

common

factors of 24

and

32.

(6)

Find the greatest

(b)

11.

How many units

is it

common from

factor of 24

3 to

-4 on

a

and

32.

number

line?

(4)

Find each missing number: 12. 6

•

6

*

z = 1224

13. $100.00

- k = $17.54

(3)

14. (3)

w

- 98 - 432

15. (3)

20x = $36.00

44

Saxon Math 8/7

16. (3)

18. f6j

w

= 200

20

17.

= 3Q

X

(3)

9 have a remainder? Does the quotient of 4554 can you tell without dividing? -e-

How

Simplify: 19. (1)

36,475 + 55,984

476

20. (i)

38

x

- $72.45

21. $80.00 (i)

22.

49 + 387 + 1579 + 98

(i)

40

23. $68.00

24. 8

25.

7

•

•

5

(i)

(i)

Compare: 4000 ^ (200 ^ 10)

Q

t

4000 + 20 °) ^ 10

12. 4}

26. Evaluate (a)

each expression

ab

(b)

a

for a

= 200 and b = 400:

- b

(

(c)^

b a

27. Refer (7)

28.

to

answer

(a)

the

figure

and

(b).

at

right

to

(a)

Which

angle

is

an acute angle?

(b)

Which

angle

is

a straight angle?

What type

of angle

is

formed by perpendicular lines?

(7)

Refer to the figure below to answer problems 29 and 30. x 29.

Name

Y

Z

three segments in this figure.

(7)

30. l7)

If

you kne w

find

mXZ.

mXY

and mYZ, describe how you would

45

Lesson 8

LESSON Fractions and Percents

Inch Ruler WARM-UP Facts Practice: 64 Multiplication Facts (Test A)

Mental Math: a.

4 - 10

b.

SO. 25

d.

325 + 50

e.

347 - 30

g.

Start

with a score. + 1,-5-3,

+

-r

3,

10

x

x

c. f.

5

+

1,

-r

- 65c

SI. 00

200 4,

+

x

10

1,

-r

2,

X 6

S

3.

Problem Solving:

The number 325 contains the

three digits 2.3. and 5. These three can be ordered in other ways to make different numbers. Each such ordering is called a permutation of the three digits. The smallest permutation of 2, 3, and 5 is 235. Which number is the largest permutation of 2. 3, and 5? digits

NEW CONCEPTS Fractions

and

percents

Fractions and percents are whole or parts of a group.

At

right

we

use

a

commonly used

whole

circle

to

name

parts of a

to

represent 1. The circle is divided into four equal parts with one part shaded. One fourth (|) of the circle is shaded, and | of the circle is not shaded.

Since the whole circle also represents 100% of the circle, we can divide 100% by 4 to find the percent of the circle that is shaded.

100%

We is

find that

25%

-r

of the circle

4 = is

25%

shaded, so

75%

of the circle

not shaded. written with two numbers and a number below the bar is the denominator

A common

fraction

division bar.

The

is

—

1

Saxon Math 8/7

and shows how many equal parts are in the whole. The number above the bar is the numerator and shows how many of the parts have been selected. numerator

denominator

— —

—

-*

4

*-

—

division bar

A

percent describes a whole as though there were 100 parts, even though the whole may not actually contain 100 parts. Thus the "denominator" of a percent is always 100.

means 25 percent F

—25 100

we

Instead of writing the denominator, 100, percent or the percent symbol, %.

A whole number plus

use the word

mixed number. To name the number of circles shaded below, we use the mixed number 2|. We see that 2| means 2 + f To read a mixed number, we first say the whole number; then we say "and"; a fraction

is

a

.

then

we

say the fraction.

Two and

three fourths

possible to have percents greater than 100%. to write 2| as a percent, we would write 275%. It is

Example

1

Solution

If

we were

Name

the shaded part of the circle as a fraction and as a percent.

Two

of the five equal parts are shaded, so the fraction that

shaded

is §.

Since the whole circle (100%) each part is 20%.

100%

Two

is

parts are shaded.

So

-r

2 x

is

divided into five equal parts,

5

=

20%

20%,

or

40%,

is

shaded.

Lesson

Example

2

Which

47

of the following could describe

portion of this rectangle that

the

8

is

shaded? A. I

Solution

40%

B.

C.

60%

shaded and an unshaded part of this rectangle, but the parts are not equal. More than | of the rectangle is shaded, so the answer is not A. Half of a whole is 50%. There

is

a

100%

50%

- 2 =

Since more than 50% of the rectangle choice must be C. 60%.

is

shaded, the correct

Between the points on a number line that represent whole numbers are many points that represent fractions and mixed numbers. To identify the fraction or mixed number associated with a point on a number line, it is first necessary to discover the number of segments into which each length has been divided. Example

3

Point

A

represents

what mixed number on A

Solution

We

8

number

line?

—

7

this

9

A

represents a number greater than 8 but less than 9. It represents 8 plus a fraction. To find the fraction, we first notice that the segment from 8 to 9 has been divided into five smaller segments. The distance from 8 to point A crosses two of the five segments. Thus, point A see that point

represents the

Xote:

mixed number

8i.

on the number of segments and not on the number of vertical tick marks. The four vertical tick marks divide the space between 8 and 9 into five segments, just as four cuts divide a candy bar It is

important

to focus

into five pieces.

Inch ruler

A

ruler

is

a practical application of a

number

line.

The

units

standard length and are often divided successively in half. That is. inches are divided in half to show half inches. Then half inches are divided in half to show quarter inches. The divisions may continue in order to

on

a ruler are of a

48

Saxon Math 8/7

show

eighths,

sixteenths,

thirty-seconds,

and even

sixty-

fourths of an inch. In this book we will practice measuring and drawing segments to the nearest sixteenth of an inch.

we show

view of an inch ruler with one sixteenth of an inch. We have labeled each

Here

a magnified

divisions to division for reference.

16

18,83s 13 16

11

1

1

16

8

16

4

3

16

16

5

16

1

Z

T 15 16

4

2

inch

important to bear in mind that all measurements are approximate. The quality of a measurement depends upon many conditions, including the care taken in performing the It

is

measurement and the precision of the measuring instrument. The finer the gradations are on the instrument, the more precise the measurement can be. For example, if we measure segments AB and CD below with an undivided inch ruler, we would describe both segments as being about 3 inches long. B

A •-

T

inch

1

2

3

C

D

•

•

inch

We

-•

1

2

3

the measure of each segment is 3 inches ± \ inch ("three inches plus or minus one half inch"). This means each segment is within \ inch of being 3 inches long. In fact, for any measuring instrument, the greatest possible error due to the instrument is one half of the unit that marks the instrument.

We

can

say

that

can improve the precision of measurement and reduce the possible error by using an instrument with smaller units. Below we use a ruler divided into quarter inches. We see that AB is about 3\ inches and CD is about 2| inches. These

1

1

1

1

Lesson 8

49

measures are precise to the nearest quarter inch. The greatest possible error due to the measuring instrument is one eighth of an inch, which is half of the unit used for the measure.

D

c

inch

Example 4

Use an inch

AC

AB, BC, and

ruler to find

to the nearest

sixteenth of an inch.

Solution

a

b

c

•

•

•—

From point A we

AB

find

We

and AC.

measure from the

AB

center of one dot to the center of the other dot. inches, and is about 2\ inches.

AC

is

about |

B 1

1

1

1

1

1

1 1

i

i

i

|

i

i

i

|

i

1

i

| I

|

|

1

1

1

1

1

1

I

I

1

1

1

1 1

1

inch

"I

2

1

We move the zero mark on the ruler to point B to measure BC. We find BC is about l| inches. B

——•

• 1

1

1

1 1

1

1

1

1

I

'

inch

I

'

' I

"

I

'

1

I

|

I

I

|

I

I

|

|

I

I

2

LESSON PRACTICE Practice set

a.

What

fraction of this circle

is

not

is

not

shaded? b.

What percent

of this circle

shaded?

c.

Half of a whole

is

M

|

what percent of the whole?

)

— 50

—

Saxon Math 8/7

Draw and shade

circles to illustrate

each fraction, mixed

number, or percent: d.

75%

e.

f

f.

Points g and h represent what

number

mixed numbers on these

lines?

—«

1_

XZ to

1

the nearest sixteenth of an inch.

x

——• j.

I

14

13

12

Find

—

9

—

-

i.

2f

y

z

•

•

Jack's ruler is divided into eighths of

*-

an inch. Assuming

used correctly, what is the greatest possible measurement error that can be made with Jack's ruler? Express your answer as a fraction of an inch. the ruler

is

MIXED PRACTICE Problem set

1. (4,8)

2. 181

3. (1 '

Use

digits

and

tj^gg fourths

is

comparison symbol to write "One and greater than one and three fifths." a

Refer to practice problem and YZ.

What

is

the quotient

divided by the

4. List

sum

above. Use a ruler to find

i

when

of 10

the product of 20

XY

and 20

is

and 10?

the single-digit divisors of 1680.

(6)

5.

w

Point

A

represents

what mixed number on

—(—

6.

number

line?

H

(2,

this

(a)

4-

Replace the circle with the proper comparison symbol.

4)

3 (b)

What property comparison?

+

202

of addition

+ 3 is

illustrated

by

this

Lesson 8

7.

Use words

to write

51

32500000000. *

(5)

8.

What

(a)

fraction of the circle

is

shaded?

What

(b)

fraction of the circle

is

not shaded?

9.

What percent

(a)

w

of the rectangle

is

of the rectangle

is

shaded?

What percent

(b)

not shaded?

10. (8)

What is the name of the part of a fraction the number of equal parts in the whole?

that indicates

Find each missing number: 11. a

- S4.70 = S2.35

12. b + S25.48 = S60.00

(3)

13.

(3)

8c = S60.00

14. 10.000

(3)

15. (3

- d = 5420

(3)

— 15

17. 8

= 15

=

16.

f

(3)

+ 9 + 8 + 8 + 9 + 8 + n = 60

(3)

Simplify: 18. (1)

x

400 500

20.

3625 + 431 + 687

21.

6000 + 50

19. (1)

22.

24

23.

18

fU

25.

If

t

is

(a) t

x

20-10-5 3456

.

id

1000 and vis

- v

79c 30

6

11, find (b)

v -

f

52

Saxon Math 8/7

26.

The

{2]

is

rule of the following sequence

is

k = 3n -

1.

What

the tenth term of the sequence?

2,5,8,11,... 27. (2.

Compare: 416 - (86 + 119)

Q (416 -

86) + 119

4)

Refer to the figure at right to answer

problems 28 and 28.

Name

A

D

29.

the acute, obtuse, and right

angles.

29.

(a)

Name

a

segment parallel

(b)

Name

a

segment perpendicular

to

DA.

(7)

30. Referring to the figure below, l7)

meaning between the notations Q

to

wha t

DA. is

the difference in

QR and QR? R

S

53

Lesson 9

LESSON Adding, Subtracting, and Multiplying Fractions • Reciprocals WARM-UP Facts Practice: 30 Equations (Test B)

Mental Math:

-

a.

3

5

b.

$0.39 x 10

d.

342 + 200

e.

580 - 40

g.

Start

with half a dozen, +

x 6,

1,

C.

—

2,

-f

$1.00 - 290

f.

500

2,

+

4,

50

-r

•4,-5,

"T

x 15.

Problem Solving: Find the next four numbers in

this sequence:

113

1

16' 8' 16' 4'

•'•

NEW CONCEPTS Adding fractions

On

the line below,

AC by measuring

AB

is

if

by adding if

or

BC is

and

in.

in.

if in. and if in.

We

can find

B

A 3 1g,n.

i

S

18

•

.

in.

1-,n.

4

+ 1^

7

.

in.

8

.

= 21- m. 8

adding fractions that have the same denominators, we add the numerators and write the sum over the common denominator.

When

Example

1

Find each sum: v (a)

3 2 1 + - + -

(b)

33±%

+

33^%

(a)

^ + ! + ! =

(b)

33±%

+

33^% = 66|%

,

Solution

O

O

«J

54

Saxon Math 8/7

When

and denominator

the numerator

of a fraction are equal (but not zero),

the

fraction

equal

is

shows | one whole circle.

illustration is

Example 2

Solution

fractions

Example

3

- + - = - = 5

To

fractions

1

5

subtract

a

fraction

Find each difference: 3 >

Multiplying

of a circle,

The which

from a fraction with the same denominator, we write the difference of the numerators over the common denominator.

(a

Solution

1.

Add: f + % 5 5

5

Subtracting

to

(a)

The

-

i

1

w

!

3- - 1- = 2-

(b)

9

9

first

illustration

circle.

9

The second

i - i - — - = — = 5

shows

illustration

5

5

of a

\

shows

We

|

of | of a circle. see that | of \ is |. translate the word of into a multiplication symbol and find 1 U1 of ^

We

2

1

1

2

2

0ff

2

by multiplying: \ of \ becomes \ x \ = \

To multiply

we

multiply the numerators to find the numerator of the product, and we multiply the denominators to find the denominator of the product. Notice that the product of two positive fractions less than 1 is less than either fractions,

fraction.

Example 4

Find each product: (a)

Solution

(a)

\ of \ 1 1 x -

(b) y J

2

1 (b)

6

•

4

•

5 3^

40

.

Lesson 9

Reciprocals

If

we

Note

by switching the numerator and

invert a fraction

denominator,

we form

the reciprocal of the fraction.

The reciprocal

of |

is |.

The reciprocal

of |

is |.

The reciprocal

of \

The reciprocal

of 4 (or

this very

4

12 12

1

4

4

4

1

4

3

Solution

Solution

(a)

|

(a)

The

Solution

its

reciprocal

is 1,

1

The

3

reciprocal of § is f . reciprocal of

,

which

3,

is 3

3

Find the missing number: —n =

"wholes" or

y, is 3.

1

The expression |n means "| times n." Since the product of number must be the reciprocal of f, I and n is 1, the missing which is |. 4

7

3 "

(b)

3.i Example

is 4.

Find the reciprocal of each number:

(b)

Example 6

which

y) is \.

and

of a fraction

4

5

is y,

important property of reciprocals:

The product

Example

55

3

= 12

=1

How many |'s

are in 1?

The answer

the reciprocal of f

is

Lesson 2 commutative

In

we

check

12

,

which

is

f

noted that although multiplication is (6x3 = 3x6), division is not commutative

56

Saxon Math 8/7

(6

*

3

-r

3

Now we

6).

-i-

can say that reversing the order of

division results in the reciprocal quotient.

6-^3

= 2

3,6=1 LESSON PRACTICE Practice set

Simplify: o a.

5 1 + —

3 -

x

4 —

2 -

x

f r.

3

7

g.

d.

4

2

4

3

5

5

,332

13

- x —

5

e.

b.

6

6

c.

i

14|% + 14|%

h.

- + - + 3

3

5

5

8

8

3

87|% - 12|%

Write the reciprocal of each number: 4 i.

•

8

r

7 7

j.

5

k. 5

Find each missing number: 1.

fa =

m.

1

n. Gia's ruler is

6m

=

1

What

divided into tenths of an inch.

of an inch represents the greatest possible

due

error o.

to Gia's ruler?

How many |'s

p. If a

-r

fraction

measurement

Why?

are in 1?

b equals

4,

what does b + a equal?

MIXED PRACTICE Problem set

1. (1)

2. 111

What

is

the quotient

when

divided by the product of

the

1, 2,

sum

of

1,

2,

and

3 is

and 3?

The sign shown is incorrect. Show two ways to correct the sign.

Apples 0.450 per

—

pound

Lesson 9

3. ,g)

Replace each circle with the proper comparison symbol. Then write the comparison as a complete sentence, using words to write the numbers. (a)

4.

57

M-2Q-4

\0\-\

Write twenty-six thousand in expanded notation.

(5)

5.

(a)

A

dime

is

what

(b)

A

dime

is

what percent of a

(a)

What

fraction of a dollar?

(8)

6. (8}

dollar?

fraction of the square

is

shaded? (b)

What

fraction of the square

is

not shaded? 7. ,7)

8.

w

an imaginary "line" from the Earth ray, or a segment? Why? Is

Use an inch

ruler to find

LM, MN, and

Moon

LN to

M

(a)

List the factors of 18.

(b)

List the factors of 24.

(c)

Which numbers

(d)

Which number

N

(6)

10. If (1,

n

is |,

are factors of both 18

is

the

GCF

of 18

find

9)

(a)

and 24?

and 24?

n + n

(b)

n - n

Find each missing number: 11. 85,000

+ b = 200,000

13.

d + $5.60 = $20.00

(3)

14. e x 12 = $30.00 (3)

15. (3)

12. O)

(3)

/ - $98.03 = $12.47

900 + c = 60

a line, a

the nearest

sixteenth of an inch. L

9.

to the

58

Saxon Math 8/7

+ 5 + 7 +

16. 5 + 7

.6

+ u +

l

+ 2 +

3+

1

4

8

8

4 = 40

(3)

Simplify: 17.

11 3—

9)

15

(

19.

-

O)

4

1 1—

-

18.

1- + l|

15

x

-

20.

1802 17

4

21. $8.97 +

$110 + 530

(i)

22. (1)

$60.00 - $49.49

25. 50

in

4 —

(9j

5

x

2 -

x

1 -

3

27.

3

(9)

28. Refer to the figure at right to

60

•

70

- + - + 9

9

9

answer

(a)and(b). (a)

Which

(b)

Which segment to

angles are acute? is

perpendicular

CB?

29.

Use words

(2,8)

the next

to describe the following sequence.

number

in the sequence. i

J->

(9)

•

(i)

« 26.

30.

78

x

24. $0.09 x 56

(7)

607

23.

How many |'s

are in 1?

I 1 1

•*• 2' 4' 8'

Then

find

Lesson 10

59

LESSON Writing Division Answers as

Mixed Numbers

Improper

•

Fractions WARM-UP Facts Practice: 64 Multiplication Facts (Test A)

Mental Math: - 10

a.

7

d.

384 + 110

g.

3 x 6,

-r

x

2,

5,

10

b.

SI. 25

e.

649 - 200

+

3,

4-

6,

x

-

3,

SI. 00

c.

4,

+

1,

82(2

300 4 30

f.

x

-

4-

3

Problem Solving:

Copy

this

problem and

fill

_37_ 2_65

in the missing digits:

59_7

NEW CONCEPTS Writing division

answers as mixed

numbers

Alexis cut a 25-inch ribbon into four equal lengths. was each piece?

To

However, remainder does not answer the

find the answer to this question,

expressing the answer with a

we

How long

divide.

question.

_6R

1

4l25 24 1

The answer 6 R 1 means that each of the four pieces of ribbon was 6 inches long and that a piece remained that was 1 inch long. But that would make five pieces of ribbon! Instead of writing the answer with a remainder, the answer as a

we

will write

mixed number. The remainder becomes the

Saxon Math 8/7

numerator of the denominator.

and we use the divisor

fraction,

as the

4J25 24 1

This answer means that each piece of ribbon was 6| inches long,

Example

1

Solution

which

is

What percent

One

the correct answer to the question. of the circle

third of the circle

is

is

shaded?

shaded, so

we

divide

100% by

3.

33|% 3)100% _9 10 _9 1

We find that 33f % Improper fractions

A

of the circle

is

shaded.

equal to 1 if the numerator and denominator are equal (and are not zero). Here we show four fractions equal to 1. fraction

is

2 2

A

4 4

3 3

5

5

than 1 is called an improper fraction. Improper fractions can be rewritten either as whole numbers or as mixed numbers.

Example 2

fraction that

is

equal to

1 or is greater

Convert each improper fraction to either a whole number or a mixed number: (a)

§

(b)

§

Lesson 10

Solution

(a)

Since | equals

1,

the fraction f 5

2

3

3

i+

-

(b)

Likewise, |

is

greater than

is

greater than

61

1.

f

1.

.33

6 3

3

3

=

+

1

1

= 2

We

can find the whole number within an improper fraction by performing the division indicated by the fraction bar. If there is a remainder, it becomes the numerator of a fraction whose denominator is the same as the denominator in the original improper fraction. 2

1

3j5

(a)

3

(

3]6~

b )|

3 2

This picture illustrates that f is equivalent to l|. By shading the remaining section we could illustrate that § equals 2.

Example

3

Solution

Rewrite 3| with a proper fraction.

The mixed number 3| means

3

+

1.

to if.

5

5

Now we

combine

3

and

l|.

3

+

i

- 4!

The

fraction ? converts

Saxon Math 8/7

When

the answer to an arithmetic problem is an improper fraction, we may convert the improper fraction to a mixed

number.

Example 4

Simplify: (a)} 1

Solution

(a)

* +

i

5

5

-

(b) K }

^

x

4

2

%5 + |5 = |5 = if5

(b)

5

= 15 = lZ

3

x

4

2

8

8

Sometimes we need to convert a mixed number to an improper fraction. The illustration below shows 3| converted to the improper fraction

13

1

We see that every whole circle equals \. is

Example

5

Solution

| + |

+

|,

which equals ^. Adding

So three whole \

more

totals

circles

^.

Write each mixed number as an improper fraction: (a)

3

(a)

The denominator

(b)

(b)

:

The denominator

is 3,

so

3

3

+

3

is 4,

so

4

§,

we

use | for

3

+

3

use | for

1.

4 4

3

4

Thus 3|

is

Thus 2§

is

= 10

3

3

we

l

1.

uf

11 4

we

use | for 1. If we multiply find that 12 equals ^. Thus, 12| is

The denominator 12 by

we +

3

4

(c)

(0

2|

is 2,

12 1^1 +

so

24

1

25

2

2

2

.

63

Lesson 10

The solution to example 5(c) suggests a quick way to convert a mixed number to an improper fraction. Multiply the denominator of the fraction by the whole number, add the numerator of the fraction, and put the result over the denominator of the

fraction. So, for 12|

we have

+

^1

1

1Z

_ 19 LZ 2 "

^

2

_ 2 x 12 + ~ 2

24 + 1

1

"

25

"

2

2

x

LESSON PRACTICE Practice set

a.

Alexis cut a 35-inch ribbon into four equal lengths. long was each piece?

b.

One day

is

How

what percent of one week?

Convert each improper fraction to either a whole number or a

mixed number: 11

c

f.

11

d

Draw and shade

e

2^

circles to illustrate that 2\ 4

= f 4

Simplify: 2

8

*

3

+

2

+

3

2

7

,

h '

3

3

X

2

.

3

,2 21

,2 3

Convert each mixed number to an improper fraction: j.

k. 3g

1-^

m. 5|

n.

1.

o.

6f

4^

lOf

MIXED PRACTICE Problem set

1. {2,

91

2. 171

\ to write an equation that illustrates the associative property of multiplication.

Use the

fractions

\,

|,

and

Use the words perpendicular and parallel to complete the following sentence: a rectangle, opposite sides are adjacent sides are ( p )

In

.

(a)

and

.

64

Saxon Math 8/7

3. (l1

4.

What

is

the difference

when

the

subtracted from the product of

What percent

(a)

(8)

sum

is

of the rectangle

is

2,

3,

and 4

is

and 4?

2, 3,

of the rectangle

of

shaded?

What percent

(b)

not shaded? 5.

Write 3f as an improper fraction.

(10)

6.

Replace each circle with the proper comparison symbol:

(4.9)

7. 181

-

2

(a)

Point

2

O

2

+ 2 2

2

w

2

2

M represents what mixed number on this number

line?

M i

+

I

1

1

10

8.

Draw and shade

circles to

show

do)

9. List

the single-digit

numbers

11

that l|b = fb

that are divisors of 420.

(6)

Find each missing number: 10. 12,500

+ x = 36,275

11.

(3)

18y = 396

(3)

12. 77,000

- z = 39,400

13. (3)

(3)

14. b

- $16.25 = $8.75

- =

15. c

(3)

$1.25

8

+ $37.50 = $75.00

(3)

16. 8

+ 7 + 5 + 6 + 4 +

17

+

3

+ 7 = 50

(3)

Simplify:

^ 17.

-

r«y

2

20.

5

x

5 -

4

18 f9j

2000 - (680 - 59)

(i)

8

19

8

21.

100%

+ 9

(10)

(2)

22. 89(2 + 57(2

— — —

+ $15.74

23. a)

800 x 300

^ + ^ 20 20

65

Lesson 10

24. fjoj

2

2

2- + 23

3

26. Describe {7}

each figure as a

symbol and

letters to

25.

2 -

o)

3

•

2 3

line, ray, or

name each

•

2 3

segment. Then use a

figure.

H

27.

How many £'s

are in 1?

(9)

28.

What

are the next three

numbers in

this

sequence?

(2, 8)

32, 16, 8, 4, 2,

29.

Which

of these

numbers

is

...

not an integer?

(4)

A. -1 30. (a) If

B.

a

— b =

5,

C. \

what does b - a equal?

(4,9)

(b)

If

^

=

3,

what does ~ equal?

D. 1

66

I

Saxon Math 8/7

Focus on

Investigating Fractions

and

Percents with Manipulatives In this investigation students will

make

a set of fraction

manipulatives to use in solving problems with fractions. Materials needed: •

Photocopies of Activity Masters 1-6 (1 copy of each master per two students; masters available in the Saxon Math 8/7 Assessments and Classroom Masters).

•

Scissors

•

Envelopes or locking plastic bags (optional)

Note: Color-coding the fraction manipulatives makes sorting easier. If you wish to color-code the manipulatives, photocopy each master on a different color of construction paper. Following the activity, each student may store the fraction manipulatives in an envelope or plastic bag for use in later lessons. Preparation:

Have students separate the fraction manipulatives by cutting out the circles and cutting apart the Distribute

materials.

fraction slices along the lines.

Activity:

Working

Using Fraction Manipulatives

groups of two or three, use your fraction manipulatives to help you with the following exercises: in

1.

What

fraction

is

half of f ?

2.

What

fraction

is

half of |?

3.

What

fraction

is

half of f ?

4.

What

fraction

is

half of

6

Investigation

1

67

5.

What

fraction

is

| of §?

6.

What

fraction

is

§

7.

How many twelfths

8.

Find a single fraction piece that equals ^.

9.

Find

a single fraction piece that equals |.

10.

Find

a single fraction piece that equals

11.

How many sixths

12.

How many twelfths

13.

With a partner, assemble five | pieces to illustrate a mixed number. Draw a picture of your work. Then write an equation that relates the improper fraction to the mixed number.

14.

Find

15.

With

assemble nine | pieces to form f of a circle and I of a circle. Then demonstrate the addition of b and recombining the pieces to make \\ circles. by | | Draw a picture to illustrate your work.

16.

Two

off? equal |?

equal |?

equal |?

a single fraction piece that equals |.

a partner,

form half of a circle. Which two different manipulative pieces also form half of a circle? \ pieces

Find a fraction

to

17.

| + | + a=

19.

|+

c =

complete each equation: 1

18.

I + b =

20.\

|

+ d =

Find each percent: 21.

What percent

of a circle

is

22.

What percent

of a circle

is

f

of a circle?

^ of a circle:

\

\

23.

What percent

of a circle

is

§ of a circle?

24.

What percent

of a circle

is

| of a circle?

25.

What percent

of a circle

is

26.

demonstrate the subtraction 1 write the answer.

27.

What

Use four

\ + y^?

|'s to

fraction piece,

when used

|,

and

twice, will cover | of a

circle?

28.

What

fraction piece,

when used

three times, will cover |

of a circle? 29. If

you subtract

^

of a circle from | of a circle,

fraction of the circle 30.

Find as

many ways

what

is left?

as

you can

to

make

half of a circle

using two or more of the fraction manipulative pieces. Write an equation for each way you find. For example, \ - \ and § + f = §•

Extension

Write

new problems

for other

groups to answer.

Lesson

11

69

.ESSON 11

Problems About Combining Problems About Separating

(VARM-UP

Facts Practice: 30 Improper Fractions and

Mixed Numbers

(Test C)

Mental Math: a.

$7.50 +

75(2

b.

$40.00

d.

(3 x 20)

+ (3x5)

e.

250 - 1000

g.

Start

with the number of hours in a day, +

+

-r

4,

-r

10

c. f.

$10.00 - $5.50 \ of 28 2,

x

3,

-r

4, x 5,

7.

Problem Solving: Letha has 7 coins in her hands totaling 50c. What are the coins?

JEW CONCEPTS In this lesson

we

will begin solving one-step story problems

writing and solving appropriate equations for the problems. To write an equation, it is helpful to understand the plot of the story. All stories with the same plot can be modeled with the same equation, which is why we say they follow a pattern. There are only a small number of story problem plots. In this lesson we deal with two of them.

by

Problems about combining

One common Here

is

idea in story problems

is

that of combining.

an example of a complete story about combining: Albert has $12. Betty has $15. Together they have $27.

Stories like this

have an addition pattern.

some + some more = S +

M=

total

T

There are three numbers in this pattern. In a story problem one of the numbers is missing. To write an equation, we use a letter to stand for the missing number. If the total is missing, we add to find the missing number. If an addend is missing,

70

Saxon Math 8/7

known addend from the sum to find the missing addend. Although we sometimes use subtraction to

we

subtract the

important to recognize that story problems about combining have addition thought patterns. solve the problem,

We

follow four steps

1

when

solving story problems:

1:

Read the problem and identify

Step

2:

Write an equation for the given information.

Step

3:

Solve and check the equation.

Step

4:

Review the question and write the answer. follow these steps as

we

its

pattern.

consider some examples.

At the end of the first day of camp, Marissa counted 47 mosquito bites. The next morning she counted 114 mosquito bites.

Solution

is

Step

We will Example

it

How many new bites

Step

1:

We

Step

2:

We

did she get during the night?

recognize that this problem has an addition pattern. Marissa had some mosquito bites, and then she got some more. write an equation for the given information.

Marissa had 47 bites. She got some more she had a total of 114 bites.

47 + Step

3:

We

M

bites.

Then

= 114

M, the missing number in the equation. To find a missing addend, we subtract. Then we check our work by substituting the answer into the original find

equation.

Step

Example 2

4:

114 47

47 bites + 67 bites

67

114 bites

check

Now we

review the question and write the answer that completes the story. During the night Marissa got 67 new bites.

scout troop encamped in the ravine. A second troop of 137 scouts joined them, making a total of 312 scouts. How many scouts were in the first troop?

The

first

Lesson

Solution

Step

1:

Step

2:

71

1 1

We

recognize that this problem is about combining. It has an addition pattern. There were some scouts. Then some more scouts came.

We

write an equation using S to stand for the

number

of scouts in the first troop.

S + 137 = 312 Step

3:

We

find S. the missing

find a missing addend,

number

we

in the equation.

subtract.

To

Then we check

our work by substituting into the original equation.

Step

Problems about separating

4:

312 - 137

+ 137 scouts

175

312 scouts

1

/

5

scouts

check

Now we

review the question and write the answer. There were 175 scouts in the first troop.

Another

common

amount

into

idea in story problems

is

separating an

two parts. Often problems about separating involve something "going away." Here is an example:

Ted wrote a check to Xed for S37.50. If S824.00 was available in Ted's account before he wrote the check, how much was available after he gave the check to Xed? Story problems like this have a subtraction pattern.

beginning amount - some went away = what remains

B - A

=

R

There are three numbers in this pattern. In a story problem one of the three numbers is missing. To write an equation, we use a letter to represent the missing number. Then we find the missing number and answer the question in the problem.

We

follow the same four steps problems with addition patterns.

Example

3

we

followed in solving

dozen cookies. While they were cooling, he went to answer the phone. When he came back, only 32 cookies remained. His dog was nearby, licking her chops. How many cookies did the dog eat while Tim was answering the phone?

Tim baked

4

Saxon Math 8/7

Solution

Step

1:

We

recognize that this problem has a subtraction

Tim had some

pattern.

Step

2:

cookies.

We

write an equation using 48 for 4 dozen cookies and A for the number of cookies that went away.

A

48 Step

3:

Then some went away.

We

= 32

find the missing number.

in a subtraction pattern,

we

To

find the subtrahend

subtract.

Then we check

our work by substituting the answer into the original equation.

48

- 32

48 cookies

—-

- 16 cookies

i

Step

4:

check

32 cookies

16

Now we

review the question and write the answer. While Tim was answering the phone, his dog ate 16 cookies.

Example 4

The room was

full of

boxes

when Sharon

began.

shipped out 56 boxes. Only 88 boxes were boxes were in the room when Sharon began? Solution

Step

1:

left.

Then she

How many

We recognize that this problem is about separating. It has a subtraction pattern. There were boxes in a room. Then Sharon shipped some away.

Step

2:

We write an equation using B to of boxes in the

room when

B Step

3:

We

stand for the number Sharon began.

56 = 88

find the missing number.

a subtraction pattern, difference.

Then we

To

find the

minuend

in

we add

the subtrahend and the check our work by substituting

into the original equation.

88 + 56

144 Step

4:

Now we

—-

|

—

I

144 boxes - 56 boxes 88 boxes

check

review the question and write the answer. There were 144 boxes in the room when Sharon began.

Lesson

11

73

LESSON PRACTICE Practice set

Follow the four-step method shown in this lesson for each problem. Along with each answer, include the equation you used to solve the problem. a.

on the scales. Billy weighed 118 pounds. Then both Lola and Billy stood on the scales. Together they weighed 230 pounds. How much did Lola weigh?

b.

Lamar cranked

Billy stood

for a

the crank 216 turns. how many turns did

c.

number

Then Lurdes gave number of turns was 400,

of turns.

the total Lamar give the crank?

If

At dawn 254 horses were in the corral. Later that morning Tex found the gate open and saw that only 126 horses remained. How many horses got away?

d.

Cynthia had a lot of paper. After using 36 sheets for a report, only 164 sheets remained. How many sheets of paper did she have at first?

e.

Write a story problem about combining that

fits

this

fits

this

equation:

$15.00 +

f.

T =

$16.13

Write a story problem about separating that equation:

32 - S = 25

MIXED PRACTICE Problem set

1.

2. ln)

the day of the festival drew near, there were 200,000 people in the city. If the usual population of the city was 85,000, how many visitors had come to the city?

As

Syd returned from the store with $12.47. He had spent $98.03 on groceries. How much money did he have when he went to the store?

Exactly 10,000 runners began the marathon. If only 5420 "" runners finished the marathon, how many dropped out along the way? 3.

74

Saxon Math 8/7

4. (8

What

(a)

10)

-

fraction of the group

/On o

is

shaded?

/

What

(b)

fraction of the group

is

not shaded?

\

What percent

(c)

^O O U

of the group

is

not shaded?

5. (4,

Arrange these numbers in order from

(a)

least to greatest:

8)

0,-2,1

|,

j

Which

(b)

of these

numbers

is

not an integer?

A

6.

35-inch ribbon was cut into 8 equal lengths. was each piece?

7.

Use two

m (4]

8. l5)

9.

digits is less

and symbols

How

"The product of one and than the sum of one and two." to write

Subtract 89 million from 100 million. Use words to write the difference.

(a)

List the factors of 16.

(b)

List the factors of 24.

(c)

Which numbers

(d)

What

(6)

is

the

GCF

and 24?

are factors of both 16

of 16

and 24?

Find each missing number: 10.

8000 - k = 5340

11.

(3)

•

9

•

n = 720

13.

16. (2,

4)

m

$126 +

r

= 1760 = $375

(3)

13)

(3)

1320 +

(3)

12. 4

14.

long

—

= 13

15.

S

Compare: 100 -

(3)

(5

x 20)

— 40

= $25.00

O (100 -

5)

x

20

Lesson

11

75

Simplify: 17.

1- + 1-

18.

9

(io)

9

do)

135

19. (i)

22. $140.70

35

a)

a)

(9)

3

40

(i)

21. 30($1.49)

23.

3

20.

72

x

- x -

5

1

1

9

3

2

8

(9)

25. Write 3| as

an improper

1

- +

24.

8

fraction.

(10)

26. (8)

Which choice below

the best portion of the is

estimate of the rectangle that is shaded?

B.I 27.

What

are the next four

(2,8)

40%

C.

numbers

113

D.

60%

in this sequence?

1

8' 4' 8' 2'

28. Refer to the figure at right to (7)

(a)

29. (8)

(b).

(a)

Which

angles acute angles?

appear

to

be

(b)

Which

appear

to

be

angles obtuse angles?

Use an inch point

30. If (9)

and

answer

ii

B so

-r

m

ruler to

that

AB is

draw

AC 3^ inches

if inches.

equals f what does ,

long.

Now find BC.

m

-r

n equal?

On AC mark

1

76

Saxon Math 8/7

LESSON

12

Problems About Comparing Elapsed-Time Problems

WARM-UP Mixed Numbers

Facts Practice: 30 Improper Fractions and

(Test C)

Mental Math: a.

$6.50 + 60(2

c.

$10.00 - $2.50

e.

500 - 2000

g.

Start

$1.29 x 10

.

x 20) + (4 x 3)

(4

.

f of 64

with three score,

-r

2,

+

+

2,

-r

,

2, 4

2,

•

+

2,

x 2.

Problem Solving:

The diameter circular

of a

object

is

circle

across the circle through

or a

f c|X

distance

the

its

center.

%^

Find the approximate diameter of the

penny shown

at right.

200J

1

1

1

1

1

1

1 1

1

inch

1

1

1

1

T

1

M'l'l

1

1

1

1 1

1 1

n

1

2)

NEW CONCEPTS we

practiced solving problems with subtraction patterns. Those problems were about separating an amount into two parts. In this lesson we consider two other types of problems with subtraction patterns.

In the previous lesson

Problems about comparing

Another type of problem that has a subtraction pattern is about comparing. In these problems, one amount is larger and one amount is smaller. We not only have to decide which number is greater and which number is less, but also how

much greater or how much less. The number that describes how much greater or how much less is called the difference.

We write the numbers larger

in the equation in this order:

- smaller = difference L - S =

Example

1

D

During the day 1320 employees work at the toy factory. At night 897 employees work there. How many more employees work at the factory during the day than at night?

Lesson 12

Solution

Step

1:

Step

2:

77

Questions such as "How many more?" or "How many fewer?" indicate that a problem is about comparison. Therefore, this problem has a subtraction pattern. write an equation for the given information. We use the letter D in the equation to stand for the

We

difference.

1320 - 897 = Step

3:

We

find

D

missing number in the pattern by

the

subtracting.

1320 employees - 897 employees 423 employees

As expected,

the difference

is less

than the larger of

the two given numbers.

Step

Example 2 Solution

4:

We

review the question and write the answer. There are 423 more employees who work at the factory during the day than who work there at night.

The number 620,000 Step

1:

is

how much

less

than 1,000,000?

The words how much less indicate that this is a comparison problem. Therefore, it has a subtraction pattern.

Step

2:

We

write an equation using D to difference between the two numbers.

1,000,000 - 620,000 =

Step

3:

We

stand for the

D

subtract to find the missing number.

1,000,000

-

620,000 380,000

Step

4:

We

review the question and write the answer. Six hundred twenty thousand is 380,000 less than 1,000,000.

Elapsed-time

problems

Elapsed time is the length of time between two points in time. Here we use points on a ray to illustrate elapsed time. elapsed time

Time earlier

later

date

date

person's age is an example of elapsed time. Your age is the time that has elapsed since you were born until this present

A

Saxon Math 8/7

moment. By subtracting the date you were born from today's date you can find your age. Today's date

- Your

(later)

birth date (earlier)

Your age

(difference)

Elapsed-time problems are like comparison problems. They have a subtraction pattern. later

-

earlier

= difference

L - E = Notice

how

similar this

is

D

to the larger-smaller-difference

equation.

Now it is time to Example

3

Solution

solve

some problems.

How many

years were there from 1492 to 1776? (Unless otherwise specified, years are a.d.)

Step

1:

We

Step

2:

We

recognize that this is an elapsed-time problem. Therefore, it has a subtraction pattern. write an equation for the pattern. The year 1776 is later than 1492. We use Y to represent the number of years between 1492 and 1776. [Y is the difference of these two numbers.)

1776 - 1492 = Step

3:

We

Y

subtract to find the difference.

1776 - 1492 284 Step

Example 4

Solution

4:

Now we

review the question and write the answer. There were 284 years from 1492 to 1776.

Martin Luther King year was he born? Step

1:

Jr.

died in 1968

is

is

the difference of the year of his assassination

and the year of his 2:

age of 39. In what

an elapsed-time problem. Time problems have a subtraction pattern. The age at which King This

died Step

at the

We

birth.

write an equation using

B

to stand for the year of

King's birth.

1968 -

B

= 39

79

Lesson 12

Step

3:

To

we minuend. We may

find the subtrahend in a subtraction pattern,

subtract the difference from the

check our work by substituting into the original equation.

1968 39

1968 - 1929

1929

39

Step

4:

check

Now we

review the question and write the answer. Martin Luther King Jr. was born in 1929.

ESSON PRACTICE Practice set

Follow the four-step method to solve each problem. Along with each answer, include the equation you used to solve the problem. a.

The number 1,000,000,000

is

how much

greater than

25,000,000? b.

How many years were

c.

John

d.

Write a story problem about comparing that

Kennedy died year was he born? F.

there from 1215 to 1791? in 1963 at the age of 46. In

what

fits

this

fits

this

equation:

58 e.

in.

- 55

in.

=

D

Write a story problem about elapsed time that equation:

2003 i/IIXED

B =

14

PRACTICE

Problem set

1. 1111

2. (n}

Seventy-seven thousand fans filled the stadium. As the fourth quarter began, only thirty-nine thousand, four hundred remained. How many fans left before the fourth quarter began?

When

she got home, she discovered that she already had some bananas. If she now has 31 bananas, how many did she have before she went to the store?

Mary purchased 18 bananas

at

the store.

.

80

Saxon Math 8/7

3.

How many years were there

from 1066

to

1215?

(12)

4. 1121

5.

week 77,000 fans came to the stadium. Only 49,600 came the second week. How many fewer fans came to the stadium the second week? The

first

Write a story problem about separating that

fits

this

equation:

6.

What property

is

$20.00 -

C

= $7.13

illustrated

by

this equation?

(2)

1 X

1 = -

1

2

(5,

2

how much

7. u)

Twenty-three thousand

8.

Replace each circle with the proper comparison symbol:

is

less

than one

million? Use words to write the answer.

(4, 8)

(a)

9. l7>

2

Name

-

3

O -1

(b)

O|

three segments in the figure

below

in order of

length from shortest to longest.

Q

P

10.

|

Draw and shade

11. (a)

w

What

circles to

R

show

that 2\ equals f

fraction of the triangle

is

shaded? (b)

What percent

of the triangle

is

not shaded? 12.

The number 100

is

divisible

by which whole numbers?

(6)

Find each missing number: 13.

15x = 630

15.

2900 - p = 64

y - 2714 = 3601

17. 20r

16. $1.53 + q (3)

(3)

(3)

14. (3)

(3)

= 1200

18. (3)

^- = 16 14

= $5.00

Lesson 12

81

Simplify: 19.

72,112

20.

ll}

- 64,309

(1)

21.

3

| -

23.

-

(w)

2

x

9

+

5

8

22.

9

3

9

(10)

^

24. $37.20

5

9

T*

15

id

25. Divide 42,847

m

453,978 + 386,864

by 9 and express the quotient

as a

mixed

number.

26. $4.36 + $15.96 + 760 +

$35

(i)

27. (2.

Find the next three numbers in

this sequence:

113

8)

4' 2' 4' *••

28.

How manyJ |'s 3

are in 1?

(9)

if as an improper fraction, and multiply the improper fraction by |. What is the product?

29. Write {10)

30. (7,8>

Using a ruler, draw a triangle that has two perpendicular sides, one that is | in. long and one that is 1 in. long. What is the measure of the third side?

82

Saxon Math 8/7

LESSON

13

Problems About Equal Groups

WARM-UP Facts Practice: 30 Improper Fractions and

Mixed Numbers

(Test C)

Mental Math: a.

$8.00 - $0.80

d.

(5 x 30)

g.

7 x 8,

+

+

(5x3)

4,

-r

3,

+

10

b.

$25.00

e.

250 - 500

1,

-r

3,

+

-r

8,

c. f.

x

2,

-

3,

$10.00 - $6.75 | of 86 -f

3

Problem Solving: Moe, and Larry stood side by side for a picture. Then they rearranged themselves in the order Moe, Joe, and Larry and took another picture. These arrangements are two of the possible permutations of the three people Joe, Moe, and Larry. Altogether, how many permutations (arrangements) are possible?

Joe,

NEW CONCEPT We

have used both the addition pattern and the subtraction

word problems. In this lesson we will use a multiplication pattern to solve word problems. Consider this pattern to solve

problem: Juanita packed 25 marbles in each box. If she filled 32 boxes, how many marbles did she

pack in

all?

This problem has a pattern that is different from the addition pattern or subtraction pattern. This is a problem about equal groups, and it has a multiplication pattern.

number

of groups x

number

N

x

G

in each group = total

= T

T stands for total. To find the total, we multiply. To find an unknown factor, we divide. We will consider three examples. Example

1

Juanita packed 25 marbles in each box. how many marbles did she pack in all?

If

she filled 32 boxes,

83

Lesson 13

Solution

We use the Step

1:

Step

2:

four-step procedure to solve story problems.

Since each box contains the same number of marbles, this problem is about equal groups. It has a multiplication pattern.

We

write an equation using T for the total number of marbles. There were 32 groups with 25 marbles in each group.

32 x 25 =

Step

3:

To

T

find the missing product,

we

multiply the factors. The product should be greater than each factor.

Step

Example 2

4:

Movie $820.

Solution

Step

We

review the question and write the answer. Juanita packed 800 marbles in all.

tickets sold for $5 each.

How many tickets

1:

The

32 x 25

160 64

800

total ticket sales

were

were sold?

same price. This problem about equal groups of money. Therefore, it has

Each

ticket sold for the

is

a

multiplication pattern.

Step

2:

We

write an equation. In the equation

number of total was $820. the

tickets.

N Step

3:

4:

3

x $5

find a missing factor,

We

= $820

we

164 x $5 = $820

check

review the question and write the answer: 164

tickets

Example

N

for use Each ticket cost $5 and the

divide the product by the known factor. We can check our work by substituting the answer into the original equation.

To

164 5)820 Step

we

were

sold.

were delivered to the dealer by 40 trucks. Each truck carried the same number of cars. How many cars were delivered by each truck? Six hundred

new

cars

84

Saxon Math 8/7

Solution

Step

1:

An

number of cars were grouped on each truck. The word each is a clue that this problem is about equal

equal groups. This problem has a multiplication pattern.

Step

2:

We write an equation using Cto stand for the number of cars

on each

truck.

40 x Step

3:

To find work by

C

a missing factor,

= 600

we

divide.

We

check our

substituting into the original equation.

15

40)600 Step

4:

40 x 15 = 600

check

We

review the question and write the answer: 15 cars were delivered by each truck.

LESSON PRACTICE Practice set

Follow the four-step method to solve each problem. Along with each answer, include the equation you use to solve the problem.

How

a.

Beverly bought two dozen juice bars for much did she pay for all the juice bars?

b.

Johnny planted a total of 375 trees. There were 25 trees in each row he planted. How many rows of trees did he

18(2 each.

plant? c.

Every day Arnold did the same number of push-ups. If he did 1225 push-ups in one week, then how many pushups did he do each day?

d.

Write a story problem about equal groups that

fits

this

equation:

12x = $3.00

MIXED PRACTICE Problem set

1. tl2)

Ashton was 64,309. By the 1990 census, the population had increased to 72,112. The population of Ashton in 1990 was how much greater than In 1980 the population of

in 1980? 2.

Huck had five dozen night crawlers in his pockets. He was unhappy when all but 17 escaped through holes in his pockets. How many night crawlers escaped?

85

Lesson 13

3. (12)

4. 1131

5.

6. (13]

President Franklin D. Roosevelt died in office in 1945 at the ase of 63. In what year was he bom?

The beach balls were packed 12 in each case. If 75 cases were delivered, how many beach balls were there in all?

One hundred twenty

poles were needed to construct the new pier. If each truckload contained eight poles, how manv truckloads were needed?

Write a story problem about equal groups that equation. Then answer the problem.

fits

this

than the

sum

5f = S63.75

7. (1

-

12)

8.

The product of 5 and 8?

of 5

and

8

how much greater

is

(a)

Three quarters make up what fraction of a dollar?

(b)

Three quarters make up what percent of a dollar?

(a)

9.

How manv units

is it

from -5

to

+5 on the number line?

-

10. Describe

each figure as a

m svmbol and

letters to

line, ray, or

name each

segment. Then use a

figure.

(a)

11. (a)

What whole numbers

are factors of both 24

and 36?

(S)

GCF

(b)

What

is

12. (a)

What

fractions or

points (b)

the

A

of 24

and 36?

mixed numbers are represented by and B on this number line?

Find AB. B

A i

—

•

i

•

——

1

i

«~

:

S.ixon Math 8/7

Find each missing number:

1800

36


f)

3

2

22. //o/

24. ru

20.

™»

23.

:r

w

5

r 2, >.

25

27. Wiile 2'

as an

2H.

Wbal

is

x

25

10) ( )

improper

improper fraction by

75

5

5

(100 r

:

+

.'

000

"'

Compare: L000

2

.

the product of

fraction.

Wbal

.

J!

(1000

and

is

LOO)

10

-f

Then multiply

Ibe product?

Its

reciprocal?

on

Refer to

I

figure

lie?

problems 20 and 2*). '

'

Name

al

rigbi

to

answer

A

30.

Ibe obtuse, acute,

and

right

angles.

30. (a)

AB d

(7)

))

AH

I

_i

B

87

Lesson 14

LESSON

14

Problems About Parts of a Whole

WARM-UP Facts Practice: 30 Equations (Test B)

Mental Math: a.

$7.50 - 750

b.

$0.63 x 10

c.

$10.00 - $8.25

d.

(6

e.

625 - 500

g.

Start

+

2,

f.

with three dozen, * 2.

2,

+

(6x4)

x 20) +

| of 36 2,

+

2,

+

2,

-r

2.

+

2,

-

2,

Problem Solving: Terry folded a square piece of paper in half diagonally to form a triangle. Then he folded the triangle in half as shown, making a smaller triangle. With scissors Terry cut off the upper corner of the triangle. What will the paper look like when it is unfolded?

NEW CONCEPT We

remember

problems about combining have an addition pattern. Problems about parts of a whole also have an addition pattern. that

part + part =

P1 + P2 = Sometimes the Example

1

Solution

whole

W

f

parts are expressed as fractions or percents.

third of the students earned a B on the test. of the students did not earn a B on the test?

One

What

fraction

of students. We are given only the fraction of students in the whole class who earned a B on

We

are not given the

number

The notations P, and P2 mean "part one" and "part two." Variables with small letters or numbers to the lower right are called subscripted variables. A

+

subscripted variable

is

treated as though

it

were a single

letter.

Saxon Math 8/7

Pictures often help us understand problems about fractions. Here is a picture to help us visualize the problem:

the

test.

All

Step

1:

students

We recognize this problem is about part of a whole. It has an addition pattern.

Step

2:

We

write an equation for the given information.

It

though we are given only one number, |, but the drawing reminds us that the whole class of

may seem

as

students is |. We will use the subscripted variable to stand for "not B" students. 1

+

3

Step

3:

We

B

NB

3

find the missing number,

NB

,

by subtracting.

We

can check our work by substituting into the original equation.

/

|

B students

+ | not B students | total students

Step

Example 2

Solution

4:

check

We

review the question and write the answer. Of the students who took the test, | did not earn a B.

Shemp was excited that 61% of his answers were What percent of Shemp 's answers were not correct? Step

1:

Part of

Shemp 's answers were

correct,

correct.

and part were

not correct. This problem is about part of a whole. has an addition pattern. Here we show a picture:

It

incorrect

>

100%

correct

Step

2:

We

100%.

We

use

The whole

represented by c in the equation to stand for the

write an equation.

is

N

percent not correct.

61%

+

Nc

=

100%

/

89

Lesson 14

Step

3:

We

find the missing number,

Nc

,

We

by subtracting.

can check our work by substituting into the original equation.

100% - 61% = 39%

61% Step

4:

+

39%

=

100%

check

We

review the question and write the answer. Of Shemp's answers, 39% were not correct.

LESSON PRACTICE Practice set

Follow the four-step method to solve each problem. Along with each answer, include the equation you use to solve the problem. a.

b.

Only 39% of the lights were off?

Two

fifths of

What c.

were

lights

on.

What percent

of the

the pioneers did not survive the journey.

fraction of the pioneers did survive the journey?

Write a story problem about parts of a whole that

fits

this

equation:

45%

+

G = 100%

MIXED PRACTICE Problem set

1.

2. {14}

Beth fed the baby 65 grams of cereal. The baby wanted to eat 142 grams of cereal. How many additional grams of cereal did Beth need to feed the baby?

Seven tenths of the new haircut. first

3.

What

recruits did not like their first

fraction of the

new

recruits did like their

haircut?

How many years were there

from 1776

to

1789?

(12)

4.

Write a story problem that

fits

this equation:

(13)

12p = $2.40

90

Saxon Math 8/7

5. (14)

6.

24%

If

of the students earned an

A

on the

percent of the students did not earn an

Draw and shade

show

circles to

test,

what

A?

that 3| = ™.

(10)

7. 151

8.

Use

digits to write four

hundred seven million, forty-two

thousand, six hundred three.

What property

is

illustrated

by

this equation?

(2)

3

9.

•

(a)

List the

common

(b)

What

the greatest

2

•

1

=

•

factors of

40 and

72.

(6)

10. l7)

Name

is

common

factor of

three segments in the figure

40 and 72?

below

in order of

length from shortest to longest.

w 11. Describe 181

how to

the group that

find the fraction of

is

shaded.

(

OOOOOO

Find each missing number:

- 407 = 623

12. b

13.

(3)

$20 - e = $3.47

(3)

14. 7

5/ = 7070

•

15.

(3)

(3)

— 25

= 25

5

16. (3)

.

g 7

+ 295 = 1000

17. a

6

(3)

5

Simplify: 18.

g

5

do)

-

1

-

4

19. no)

2

2

o ^

4

S

^

3- + 25

6

4

20. $3.63 + $0.87 + 96(2

7

(i)

21. 5

4

•

3

•

•

2

•

8

1

5

(i)

22.

?

(9)

3

•

^ 3

•

^

23.

3

id

20

+

^ 89

Lesson 14

24. (i)

145 x 74

26. (5)(5

^

28.

25. 30(65(2) (D

+

27.

5)

9714 - 13,456

(4)

Compare: (1000 - 100) - 10

O 1000 -

(100 - 10)

(2, 4)

29.

30.

Name

each type of angle

(a)

X\

How many |'s

(b)

are in 1?

-

illustrated: .

^

(c)

91

92

Saxon Math 8/7

LESSON

15

Equivalent Fractions • Reducing Fractions, Part 1

WARM-UP Mixed Numbers

Facts Practice: 30 Improper Fractions and

(Test C)

Mental Math: a.

$3.50 + $1.75

b.

c.

$10.00 - $4.98

d. (7 x

e.

125 - 50

f.

g.

10 -

9,

+

8,

-

7,

+

6,

-

5,

+

$4.00

-f

10

(7x2)

30) +

\ of 52 4,

-

3,

+

-

2,

1

Problem Solving:

Copy

this

problem and

fill

in the missing digits:

36

6

NEW CONCEPTS Equivalent fractions

name the same number are called Here we show four equivalent fractions:

Different fractions that

equivalent fractions.

12 As we can see from the same value.

4 8

3

4

2

6

the pictures, equivalent fractions have

1

2

3

4

2

4

6

8

We can form equivalent fractions by multiplying a fraction by fractions equal to

1.

Here

fractions equivalent to

1 2

X

li 2

2

~4

we

multiply \ by f f and | to form ,

,

|:

1 2

X

3

3~6

1 2

X

%

_ 4 I

4

8

Lesson 15

Example

1

Solution

Find an equivalent fraction

for | that

of j is with a denominator of 12.

3.

2

X

3

Example

2

Solution

has a denominator of 12.

To make an equivalent we multiply by |.

The denominator

4

_8_

4

12

Find a fraction equivalent to | that has Xext find a fraction equivalent to \ with Then add the two fractions you found.

We

multiply

to \

and

\

93

a a

fraction

denominator of denominator of

6.

6.

by f and \ by 4 to find the fractions equivalent that have denominators of 6. Then we add. |

—1

x

3

—1

X

2

2 —

2

2

6

3 —

3

3

5

6

Reducing

An

fractions,

fractions.

part

1

inch

provides

ruler

The segment

another

in the figure

example below is

counting the tick marks on the ruler, several equivalent names for \ inch. |

"

1

1

1 1

-:-

We

in.

= f

sav that the fractions

see that there are

|

1

\

|

M u h

1

1 1

we

1

)

= |

in.

4b

and

each reduce to ^ lb

\. |.

=

^ in.

in.

reduce some fractions by dividing the fraction by a fraction equal to 1.

4_4 8

By dividing

|

bv

4-

'

4

equivalent inch long. By

of

=

1

2

(4-4 (8^4

we have reduced

= =

to

We

can j be reduced

i

\.

j

1)

2)

| to \.

The numbers we use when we write a fraction are called the terms of the fraction. To reduce a fraction, we divide both terms of the fraction by

4^

2

a factor of 2

8-2 "4

both terms.

^4

1

8 - 4

~ 2

4

Saxon Math 8/7

Dividing each term of | by 4 instead of by 2 results in a fraction with lower terms, since the terms of \ are lower than the terms of |. It is customary to reduce fractions to lowest terms. As we see in the next example, fractions can be reduced to lowest terms in one step by dividing the terms of the fraction by the greatest

Example 3 Solution

Reduce

|§ to

common

lowest terms.

Both 18 and 24 are divisible by

This

is

factor of the terms.

2,

so

-r

2

18

18

24

24 ^ 2

we

divide both terms by 9

12

not in lowest terms, because 9 and 12 are divisible by

9^3

9_

12 ~ 12

-f

2.

3.

3

4

3

We

could have used just one step had we noticed that the greatest common factor of 18 and 24 is 6.

Both methods are

18

18

-r

6

3

24

24

-f

6

4

One method took two

correct.

steps,

and

the other took just one step.

Example 4 Solution

Reduce

3^

to lowest terms.

To reduce a mixed number, we reduce the the whole number unchanged.

A 12

8^4 " 12

-r

fraction

and leave

2

4 ~

3

3^ 3

- 3^ J 3 12 "

Example

5

Solution

Write

y

as a

mixed number with the

fraction reduced.

There are two steps to reduce and convert Either step may be taken first.

f

=

mixed number.

Convert First

Reduce First Reduce:

to a

f

Convert: f = l\

Convert:

^

= if

Reduce: if = if

95

Lesson 15

Example 6

7 1 - -

Simplify:

y

Solution

we

First

y

subtract.

Then we reduce.

Subtract

16

7

~ 9 9

9

Example

7

Solution

Write

70%

as a

Reduce

reduced

Recall that a percent

70%

We

2

9 ^ 3

~ 3

fraction.

a fraction

is

6^3

=

with a denominator of 100. 70

100

can reduce the fraction by dividing each term by 10 10

70

100

10.

7

10

ESSON PRACTICE Practice set*

a.

Form |, 7'

b.

three equivalent fractions for | by multiplying

and

by

I,

|. 3

Find an equivalent fraction

for | that

has a denominator

of 16.

Find the number that makes the two fractions equivalent. 4

C "

e.

?

=

i

d

20

5

fl6 j.

= 9

8

?

first fraction.

Reduce each -

3

Find a fraction equivalent to | that has a denominator of 10. Next find a fraction equivalent to | with a denominator of 10. Then subtract the second fraction you found from the

r

'

fraction to lowest terms: a

JL

-

O* O'

10

4|

k.

6£

h

A 16

L

12£

5 '

12

+

5

12

„

°'

q 7 3

T^ "

^ 1

16

m. 8 1|

Perform each indicated operation and reduce the

n

11

i.

n P

result:

52 -

8

'

3

96

Saxon Math 8/7

Write each percent as a reduced fraction: q.

t.

90%

r.

75%

s.

5%

Find a fraction equivalent to § that has a denominator of 6. Subtract \ from the fraction you found and reduce the answer.

MIXED PRACTICE Problem set

1. 1121

2. lu)

Great-Grandpa celebrated his seventy-fifth birthday in 1998. In what year was he born? Austin watched the geese fly south. He counted 27 in the first flock, 38 in the second flock, and 56 in the third flock. How many geese did Austin see in all three flocks?

40%

of the eggs

3.

If

151

were cracked?

4. 131

were cracked, what fraction of the eggs

The farmer harvested 9000 bushels of grain from 60 acres. The crop produced an average of how many bushels of grain for each acre?

With a ruler, draw a segment 2| inches long. Draw a w second segment l| inches long. The first segment is how much longer than the second segment?

5.

6. l4)

7.

Use and

digits five is

and symbols

"The product of three greater than the sum of three and five." to write

List the single-digit divisors of 2100.

(6)

8.

Reduce each

fraction or

mixed number:

(15)

9. 1111

10. (15>

Find three equivalent fractions l and I

for §

by multiplying by

§

For each fraction, find an equivalent fraction that has a denominator of 20: (a)

I

(b) i

(c)

I

J

11. Refer to this figure to

answer

Lesson 15

97

whole number

or a

(a)-(c):

(7)

T

12. ll0)

(a)

Name

the line.

(b)

Name

three rays originating at point R.

(c)

Name

an acute angle.

Convert each fraction

either a

mixed number:

f

(a)

13.

to

lb)

Compare:

(11)(6 +

f

(c)

7)066

f

+ 77

(4)

Find each missing number: 14. 39 + b = 50

15.

6a = 300

(3)

(3)

16. c

- $5 = 5C

17. (3)

(3)

^- = 35 35

Write each percent as a reduced fraction: 18.

80%

19.

(15)

20.

35%

(15)

How many |'s

are in 1?

(9)

Simplify: 21.

- + -

(w)

5

23.

-

O)

3

26. ri5j

'+

5

•

' 15y

22.

5

as)

-

24.

4

as)

12

3- - 18

8

- + 4

il - J-

25.

4

fioj

27.

12

28. Evaluate

29.

-

fj5j

each expression

for a

| 6

•

-7 + | 5 5

| 3

= 4 and b =

8:

Find a fraction equal to | that has a denominator of Add the fraction to | and reduce the answer.

6.

30. Write 2§ as an improper fraction. Then multiply the 151 improper fraction by \ and reduce the product.

m

98

Saxon Math 8/7

LESSON

16

U.S. Customary System

WARM-UP Facts Practice: 40 Fractions to Reduce (Test D)

Mental Math: a.

10 - 20

d. 4 x g.

23

Start

b.

15C x 10

e.

875 - 750

with 2 score and 10, r

2,

c.

x

3,

-

3,

$1.00 -

f.

\ of \

t

9,

+

2,

18

1

20

For each fraction, find an equivalent fraction that has a

denominator of

30: (b) !

(a)

f

An

octagon has

(c)

how many more

1

sides than a pentagon?

(18)

11. (a) (7,

Draw

a triangle that has

one obtuse angle.

18)

(b)

What kind

of angles are the other

triangle?

12. (a) (8

'

15)

What percent

of the circle

is

fraction of the circle

is

shaded? (b)

What

not shaded?

two angles of the

115

Lesson 18

13. (2,

Which property

by

illustrated

is

15)

this equation?

13

3

2

6

X

3

Find each missing number: 14.

x - - = -

15.

8

(9,15)

8

(9)

16. fa 35;

| 6

m

= |

17.

yy

— 10

+

-x =

(9)

4

19.

-

-

(15 j

2

4

6

= 210

1

Simplify: 18.

5-

ns;

10

20.

25?5

W

10

21>

45

22. 21

•

(1)

23. 2(50 in. +

21

(1)

24.

75Q x 8Q

(2,

What percent

of a

pound

40

in.)

16)

8 ounces?

is

116)

25.

(a)

How many

degrees

is

\ of a circle or | of a full turn?

(b)

How many

degrees

is

\ of a circle or \ of a full turn?

(17)

26. (a)

Use a protractor

to

draw

a 135° angle.

(17)

(b)

A

135° angle angle?

is

how many

27. Refer to the triangles

below

to

degrees less than a straight

answer

(a)-(c).

(18)

D *

C

B

E

F

X

s

(a)

Which

triangle appears to be congruent to

(b)

Which

triangle

(c)

Which

angle in

is

not similar to

z

AABC?

AABC?

ADEF corresponds to ZR in ASQR?

116

Saxon Math 8/7 28. Write a fraction equal to \ with a denominator of 6 and a (15) fraction equal to | with a denominator of 6. Then add the fractions.

9 10)

2\ as an improper fraction, and multiply the improper fraction by the reciprocal of |.

30.

Use a

29. Write

8

18)

Is

ruler to

draw

a triangle with each side 1 inch long.

the triangle regular or irregular?

117

Lesson 19

LESSON

19

Perimeter

WARM-UP Facts Practice: 30 Improper Fractions and

Mixed Numbers

(Test C)

Mental Math: a.

$8.25 + $1.75

b.

$12.00

d.

7 x 32

e.

625 - 250

g.

Start

with 4 dozen,

-r

:

6,

x

10

-r

5,

+

2,

-r

6,

c.

$1.00 -

f.

\ of 120

x

7,

+

1,

76(2

-f

2,

-

1,

Problem Solving:

Bobby has 12 tickets, and Mary has 8 tickets. How many tickets should Yin give to Bobby and to Mary so that they all have the same number of tickets? Yin has 25

tickets,

NEW CONCEPT The distance around a polygon is the perimeter of the polygon. To find the perimeter of a polygon, we add the lengths of

Example

1

What

is

sides.

its

3

the perimeter of this rectangle?

cm 2

Solution

3

The opposite

sides of a rectangle are equal in length. Tracing around the

2

cm

cm

2

rectangle, our pencil travels 3 cm, then 2

cm, then

perimeter

3

What

is

3

cm

is

3

Example 2

cm, then 2 cm. Thus, the

cm

+ 2

cm

+ 3

cm

+ 2

cm

= 10

cm

cm

the perimeter of this regular

hexagon?

8

mm

cm

118

Saxon Math 8/7

Solution

Thus the

All sides of a regular polygon are equal in length.

perimeter of this hexagon

mm

8

+ 8

mm

+ 8

mm

is

+ 8

mm

+ 8

mm

+ 8

mm

= 48

mm

or

6x8 mm Example 3

Find the perimeter of

= 48

mm 8

this polygon. All

angles are right angles. Dimensions are in feet. 11

Solution

We

and b

will use the letters a

to the

unmarked

to refer

sides. Notice that the

marked

lengths of side a and the side total 11 feet.

a + 5 = 11

So side a

5 11

is

6

ft.

Also notice that the length of side b equals the total lengths of the sides

marked

8

and

4.

8 + 4 = 6

The perimeter 8

Example 4

ft

So side b

12

ft.

of the figure in feet

+ 6

The perimeter

is

ft

+ 4

ft

+ 5

of a square

ft

is

is

+ 12

48

ft.

ft

+ 11

How

ft

long

= 46 is

ft

each side of

the square?

Solution

A

square has four sides whose lengths are equal. The sum of the four lengths is 48 ft. Here are two ways to think about this problem: 1.

The sum

of

what four

identical

+

+ 2.

divide 48

ft

= 48

ft

ft

the problem the second way, we see that by 4 to find the length of each side.

12

4}48

The length

48?

4 equals 48?

4 x

we can

is

= 48

+

What number multiplied by

As we think about

addends

of each side of the square

is

12

ft.

119

Lesson 19

Example

5

Isabel for

wants

to fence

some grazing land

250

made this sketch of How many feet of wire

ft

her sheep. She

202 ft\ Pasture

her pasture. fence does she need?

150

Solution

We

add the lengths of the sides

to find

175

ft

ft

how many

feet of

fence Isabel needs.

250

We

ft

+ 175

ft

+ 150

see that Isabel needs 777

ft

ft

+ 202

ft

= 777

ft

of wire fence.

LESSON PRACTICE Practice set*

a.

What

is

the

perimeter

of

3

this 3

quadrilateral?

in

2 5

b.

What

in.

in.

in.

the perimeter of this regular

is

pentagon? 5

c.

If

each side of a regular octagon measures 12 inches, what

is its

d.

cm

What

perimeter? is

the

perimeter

of

4

this

in.

hexagon? 2

in.

10

e.

f.

MacGregor has 100

in.

wire fence that he plans to use to enclose a square garden. Each side of his garden will be how many feet long? feet of

Draw

a quadrilateral with each side f inch long. the perimeter of the quadrilateral?

What

is

MIXED PRACTICE Problem set

1. (14>

2.

One eighth of the students in the class were left-handed. What fraction of the students were not left-handed? The

theater

was

six people left

when the

horror film began. Seventybefore the movie ended. One hundred

full

twenty-four people remained. the theater

when

it

was

full?

How many

people were in

120

Saxon Math 8/7

3. (13)

The Pie King restaurant

cuts each pie into 6 slices.

restaurant served 84 pies one week.

How many

The

slices of

pie were served?

4. (1)

5.

President Lincoln began his speech, "Four score and seven years ago ..." How many years is four score and seven?

Use words

(a)

to write

18700000. 1

J

(5)

Write 874 in expanded notation.

(b)

6. 141

7. 1161

8.

Use

digits

and other symbols

to write

"Three minus

seven equals negative four."

At what temperatures on the Fahrenheit scale does water freeze and boil? Find the perimeter of this rectangle:

(19)

6

8

9.

cm

cm

Write each number as a reduced fraction or mixed number:

(15)

(a)

10.

Find a and b

to

g

(c)

complete each equivalent

3a

(15)

* = JL

(a)

(b)

36

4 11.

(b)

3|f

4 1 = 9

Draw

a regular pentagon.

What

is

4% fraction:

b A 36

(18)

12. 1181

the

name

of a polygon that has twice as

many

sides as a quadrilateral?

13. (a)

Each angle of a rectangle measures how many degrees?

(17)

(b)

The

four angles of a rectangle total

14.

The

(h

of the sequence.

9)

rule of this sequence

is

113

k =

|n.

1

•** 8' 4' 8' 2'

how many degrees?

Find the eighth term

.

121

Lesson 19

Find the missing number in each equation. 1547 = 8998

15. a +

30b = $41.10

16. (3)

(3)

= $7.36

17. $0.32c

18. $26.57 +

d = $30.10

(3)

(3)

Simplify: 19.

- + - + -

do)

3

21.

-

as)

3

23. 50

3

•

22.

7

(15)

50

25.

3- + 8

'

11

(1)

(a)

How many |'s

(b)

Use the answer

8

100 100

24.

flj

8

8

as)

^

•

3- - -

20.

3

are in 1?

(9)

number

to (a) to find the

w AB

of ^'s in

5.

Then draw BC

26.

Use your

17,81

perpendicular to AB 2 in. long. Draw a segment from po int A to point C to complete AABC. What is the length

ruler to dra

\\

in. long.

oiAC? 27. Write 3| as (9,

101

28. ll5)

29.

fraction,

and multiply

it

by the

reciprocal of f

Find a fraction equal

to \ that has a

denominator of

10.

Subtract this fraction from ^. Write the difference as a

reduced

(8,

an improper

fraction.

What percent

of a yard

is

a foot?

16)

30. (19)

What

is

hexagon? °

the

perimeter

of

10

this

in

-

4 7

in.

6

in.

in.

1

22

Saxon Math 8/7

LESSON

20

Exponents • Rectangular Area, Part 1 • Square Root

WARM-UP Facts Practice: 40 Fractions to Reduce (Test D)

Mental Math: a.

$4.75 + $2.50

b.

36C x 10

d.

5 x 43

e.

625 - 125

g. 10 x 10,

-

10,

10,

-r

+

-

1,

c. f.

$5.00 - $4.32 \ of §

10, x 10, + 10,

-r

10

Problem Solving:

Copy

this

problem and

fill

in the missing digits:

8)

_8

NEW CONCEPTS Exponents

We

remember

we

that

can show repeated addition by using

multiplication. 5

There

has the same value as

+ 5 + 5 + 5 is

also a

show repeated

way

to

show repeated

4x5

multiplication.

We

can

multiplication by using an exponent. 5

In the expression 5

4 ,

•

5

•

5

•

5

= 54

4 and the base is 5. times the base is to be used

the exponent

The exponent shows how many

is

as a factor. exponent

base

The following examples show how we read expressions with exponents, which we call exponential expressions. 4 2

5

2

"four squared" or "four to the second power"

3

"two cubed" or "two

4

10 5

"five to the fourth

to the third

power"

"ten to the fifth power"

power"

Lesson 20

To

number

of times

23

we

use

shown by

the

find the value of an expression with an exponent,

the base as a factor the

1

exponent. 5

Example

1

Solution

5

= 625

'2*

(a)

4

2

(a)

4

2

(b)

2

(c)

10 5 = 10

3

= 4

•

4 = 16

= 2

•

2 •

2

v3y

3

Simplify: 4

2

We we

2

(b)

'2*

Solution

= 5

Simplify:

(d)

Example 2

4

first

"

-

10

(c)

•

10

2

4

3

9

2

J

(d)

= 8

2

•

10

:

10

•

10 = 100,000

•

3

find the value of each exponential expression.

Then

subtract.

4

2

-

2

3

16 - 8 = 8

Example

3

Find the missing number in 2

Solution

We

3 •

this equation: 2

3

= 2n

asked to find a missing exponent. Consider the meaning of each exponent. are

2

3

2

2-2-2 We

2

2-2-2

n

= 2n

see that 2 appears as a factor 6 times. So the missing

exponent

We

•

3

is 6.

indicate units that have been multiplied. Recall that when we add or subtract measures with like units, the units do not change.

can use exponents

to

4ft + 8ft = 12ft I

The

units of the

I

I

addends are the same as the

,

units of the

sum.

1

24

Saxon Math 8/7

However, when we multiply or divide measures, the units do change. 4

The

The

x 8

ft

ft

= 32

ft

I

= 32

ft

units of the factors are not the

result of multiplying feet

•

2

same as

by

ft

the units of the product.

feet is

square

feet,

which we

2

can abbreviate sq. ft or ft Square feet are units used measure area, as we see in the next section of this lesson. .

Rectangular

The diagram below represents the

area, part

been covered with square

1

How many

1-ft

square

does

hallway that has foot on each side.

floor of a

tiles that are 1

tiles

to

it

take to cover the floor of

the hallway?

4ft

floor

1ft

tile

1ft

8ft

We

see that there are 4 rows

there are 32

1-ft

square

and

8 floor tiles in each row. So

tiles.

The floor tiles cover the area of the hallway. Area is an amount of surface. Floors, ceilings, walls, sheets of paper, and polygons all have areas. If a square is 1 foot on each side, it is a square foot. Thus the area of the hallway is 32 square feet. Other standard square units in the U.S. system include square inches, square yards,

and square miles.

important to distinguish between a unit of length and a unit of area. Units of length, such as inches or feet, are used for measuring distances, not for measuring areas. To measure area, we use units that occupy area. Square inches and square feet occupy area and are used to measure area. It is

Lesson 20

We

include the word square or the exponent designate units of area. Unit of Length

1

1

25

when we

2

Unit of Area

inch

1

square inch 1

sq. 1

in.

in.

2

Notice that the area of the rectangular hallway equals the length of the hallway times the width.

Area of a rectangle = length

We

width

often abbreviate this formula as

A Example 4

x

What

is

= lw 5

the area of this rectangle? 3

3

in.

5

Solution

The

area of the rectangle

needed

is

the

number

in.

in.

of square inches

to cover the rectangle. 5

in.

3

in.

We

can find this number by multiplying the length the width (3 in.).

Area of rectangle =

5 in.

= 15

Example

5

in.

The perimeter

of a certain square

area of the square?

is

in.

•

(5 in.)

by

3 in.

2

12 inches.

What

is

the

1

26

Saxon Math 8/7

Solution

To

find the area of the square,

we

first

need

to

know

the

length of the sides. The sides of a square are equal in length, so we divide 12 inches by 4 and find that each side is 3 inches. Then we multiply the length (3 in.) by the width (3 in.) to find the area. 3

in.

3

Area =

3 in. x 3 in.

in.

= 9

in.

2

Example 6

Dickerson Ranch is a level plot of land 4 miles square. The area of Dickerson Ranch is how many square miles?

Solution

"Four miles square" does not mean "4 square miles." A plot of land that is 4 miles square is square and has sides 4 miles long. So the area is 4

Square root

mi

x 4

mi = 16 mi 2

The

area of a square and the length of its side are related by "squaring." If we know the length of a side of a square, we

square the length to find the area.

3 units squared

is

9 square

units.

we know

the area of a square, we can find the length of a side by finding the square root of the area.

If

The square

root of 9

square units

We

is

3

units.

often indicate square root with the radical symbol, Here we show "The square root of 9 equals 3."

9=3 Example 7

Simplify: (a)

^l^2^

(b)

Vtf

V

•

Lesson 20

Solution

(a)

To

find the square root of 121

we may

ask,

1

27

"What number

multiplied by itself equals 121?" Since 10 x 10 = 100, we try 11 x 11 and find that ll 2 = 121. Therefore, V121 equals 11. (b)

Squaring and finding a square root are inverse operations, so one operation "undoes" the other operation.

V?

= V64 = 8

ESSON PRACTICE Practice set*

Use words

Then

to

show how each exponential expression

is

read.

the base and

what

find the value of each expression.

a.

43

c.

10

expression 10 3 what

d. In the

,

number

is

number

is

the exponent?

Find each missing exponent: e.

2

3 •

2

2

=

2

n

Find each square g.

root:

VlOO

h.

V400

i.

Vl?

Find the area of each rectangle: 15

m

2

k.

j.

4

"

10m

4 5

m. n.

If

is

cm

in.

the perimeter of a square

What

cm

1.

is

the area of a park that

20 cm, what

is

is its

area?

100 yards square?

f 1

28

Saxon Math 8/7

MIXED PRACTICE Problem set

1. (1S1

2. (u>

There were 628 students in 4 dormitories. Each dormitory

housed the same number of students. were housed in each dormitory?

How many students

Thirty-six bright green parrots flew away while 46 parrots remained in the tree. How many parrots were in the tree

before the 36 parrots flew away? 3. ll4}

4.

Two hundred

twenty-five of the six hundred fish in the lake were trout. How many of the fish were not trout?

Twenty-one thousand,

fifty

swarmed

in through the front

hundred seventy-two through the back door. How many swarmed

door. Forty-eight thousand, nine

swarmed

in

in through both doors? 5. {2 20> '

The

rule of the following sequence

is

k =

2

n .

Find the

sixth term of the sequence.

M

2,4,8,16,... 6.

(a)

Arrange these numbers in order from least

(4,8)

1 (b)

Which

of these

—2

numbers

i

—I

to greatest:

n

are not integers?

Which is the best estimate of how w much of this rectangle is shaded? A. 50% B. 33|% C. 25% D. 60%

7.

8. 171

Each angle of a rectangle is a angle. Which two sides perpendicular to side

right

D J

L

"1

r

are

BC? B

9.

Simplify:

(20)

/

(a)

^

\

Q

I

(b)

10'

(c)

V12

each fraction, find an equivalent fraction that has denominator of 36:

10. For (15)

(b)

(a)

11. List the factors of

|

each number:

(6)

(a)

10

(b)

7

(c)

1

a

Lesson 20

12. He,

w)

The perimeter of a certain square is inchgg } on g j s eacn s id e f the square?

1

29

How many

2 feet.

Solve each equation:

= 54

13. 36 + a

w

14.

46 -

16.

100 =

= 20

(3)

15.

5x = 60

(3)

m

+ 64

(3)

17. 5

4

5

•

2

60

o lo. 1

= 5"

7

(20)

Simplify: 19.

l| + l£

20.

9

(10)

9

rjo, is)

21.

6345

5

5

2

6

360

22.

25

X

23.

w

3

1

- -

24.

4

f

u

(10)

25. Evaluate the following expressions for r

x

a

— 717

+

77

26. (15)

Add

m

= 3 and n =

— 277

777

17

27

Find a fraction equivalent 10.

1} + 1 4 4J 10:

has a denominator of to that fraction and reduce the sum.

yq

to | that

l| as an improper fraction. Then multiply the improper fraction by \ and reduce the product.

27. Write (10, 15)

28. (2,

Which property

is

illustrated

15)

12

3*2 29,

A common

1191

(a)

What

is

the perimeter of a

(b)

What

is

the area of a

What

is

this equation?

2

6

floor tile is 12 inches square.

(19 20)

30.

by

the

common

common

perimeter

of

floor tile?

floor tile? 5

this

in.

hexagon? 8 4

in.

10

in.

in.

130

Saxon Math 8/7

INVES !9I Focus on

Cj

Using a Compass and Straightedge, Part 1 Materials needed:

A

•

Compass

•

Ruler or straightedge

•

Protractor

compass

is

a tool

used

to

draw

Compasses are forms. Here we show two forms: circles called arcs.

radius

and portions of manufactured in various circles

gauge

3 2 MjiliLlililil,),!,!

!,

pivot point

marking point

The marking point of a compass is the pencil point that draws circles and arcs. The marking point rotates around the pivot point, which is placed at the center of the desired circle or arc. The radius (plural, radii) of the circle, which is the distance from every point on the circle to the center of the circle, is set by the radius gauge. The radius gauge identifies the distance between the pivot point and the marking point of the compass.

Concentric circles

Concentric circles are two or more circles with a common center. When a pebble is dropped into a quiet pool of water, waves forming concentric circles can be seen. A bull's-eye target is another example of concentric circles.

Investigation 2

131

To draw concentric circles with a compass, we begin by swinging the compass a full turn to make one circle. Then we make additional circles using the same center, changing the radius for each

new

circle.

Common center of four

concentric circles

1.

Regular

hexagon and regular triangle

Practice drawing several concentric circles.

the sides of a regular polygon are equal in length and all the angles are equal in measure. Due to their uniform shape, regular polygons can be inscribed in circles.

Recall that

A

all

inscribed in a circle if all of its vertices are on the circle and all of the other points of the polygon are within the circle. We will inscribe a regular hexagon and a regular

polygon

is

triangle.

First

we

fix the

compass

at a

comfortable setting that will not

We

swing the compass a full turn to make a circle. Then we lift the compass without changing the radius and place the pivot point anywhere on the circle. With the pivot point on the circle, we swing a change until the project

is

finished.

small arc that intersects the circle, as

shown below.

1

32

Saxon Math 8/7

Again we

compass without changing the radius and place the pivot point at the point where the arc intersects the circle. From this location we swing another small arc that intersects the circle. We continue by moving the pivot point to where each new arc intersects the circle, until six small arcs are drawn on the circle. We find that the six small arcs are equally spaced around the circle. lift

the

hexagon, we draw line segments connecting each point where an arc intersects the circle to the next point where an arc intersects the circle.

Now,

2.

To

to inscribe a regular

Use a compass and straightedge hexagon in a circle. inscribe a regular triangle,

We

we

to inscribe a regular

will start the process over

swing the compass a full turn to make a circle. Then, without resetting the radius, we swing six small arcs around the circle. A triangle has three vertices, but there are six points around the circle where the small arcs again.

Investigation 2

intersect

the

Therefore,

inscribe

1

33

regular triangle, we draw segments between every other point of intersection. In other words, we skip one point of intersection for each side of the triangle.

3.

Use your

With

circle.

to

a

tools to inscribe a regular triangle in a circle.

we can measure each

angle of the triangle. Since the vertex of each angle is on the circle and the angle opens to the interior of the circle, the angle is called an inscribed angle. a protractor

4.

What

5.

What

measure

each inscribed angle? (If necessary, extend the rays of each angle to perform the measurements.) is

is

the

the

sum

of

of the measures of all three angles of the

triangle?

6.

What shape

will

we make

if

we now draw segments

between the remaining three points of intersection?

Dividing a circle into

sectors

We

can use a compass and straightedge to divide a circle into equal parts. First we swing the compass a full turn to make a circle. Next we draw a segment across the circle through the center of the circle. A segment with both endpoints on a circle is a chord. The longest chord of a circle passes through the center and is called a diameter of the circle. Notice that a diameter equals two radii. Thus the length of a diameter of a

1

34

Saxon Math 8/7

twice the length of a radius of the circle. The circumference is the distance around the circle and is determined by the length of the radius and diameter, as we circle

is

will see in a later lesson.

A

diameter divides a circle into two half circles called

semicircles.

To divide used

a circle into thirds,

to inscribe a

hexagon.

we

We

begin with the process

draw

a circle

and swing

we six

small arcs. Then we draw three segments from the center of the circle to every other point where an arc intersects the circle. These segments divide the circle into three congruent sectors. A sector of a circle is a region bounded by an arc of the circle and two of its radii. A model of a sector is a slice of pie.

7.

Use a compass and straightedge

to

draw

a circle

and

to

divide the circle into thirds.

The segments we drew from the center Each angle

to the circle

formed

vertex at the center of the circle is a central angle. We can measure a central angle with a protractor. We may extend the rays of the central angle if angles.

that has

its

necessary in order to use the protractor.

Investigation 2

8.

What

9.

Each sector of a

1

35

the measure of each central angle of a circle divided into thirds? is

circle divided into thirds

occupies what

percent of the area of the whole circle?

To divide

we

again begin with the process we used to inscribe a hexagon. We divide the circle by drawing a segment from the center of the circle to the point of intersection of each small arc. a circle into sixths,

7

10.

What

is

the measure of each central angle of a circle

divided into sixths? 11.

Each sector of a

circle divided into sixths occupies

what

percent of the area of the whole circle? In problems 12-24

we

provide definitions of terms presented in this investigation. Find the term for each definition: 12.

The distance around

13.

The distance across

14.

The distance from the center

a circle

a circle through

its

center

of a circle to every point on

the circle 15. Part of the circumference of a circle

16.

A region bounded by an arc

17.

Two

or

more

circles

of a circle

and two

with the same center

radii

1

36

Saxon Math 8/7

18.

A

19.

A polygon whose vertices

segment that passes through the has both endpoints on the circle are

on a

interior of a circle

circle

and

and whose other

points are inside the circle 20.

A half circle

21.

An

22.

The distance between the pivot point and the marking point of a compass when drawing a circle

23.

The point

angle

whose vertex

that

is

the

is

the center of a circle

same distance from any point on

a

circle

24.

An

angle that opens to the interior of the circle from a vertex on the circle

The following paragraphs summarize important

facts

about

A

copy of this summary is available as Facts Practice Test E in the Saxon Math 8/7 Assessments and Classroom circles.

Masters.

The distance around a circle is its circumference. Every point on the circle is the same distance from the center of the circle. The distance from the center to a point on the circle is the radius. The distance across the circle through its center is the diameter, which equals two radii. A diameter divides a circle into two half circles called semicircles. A diameter, as well as any other segment between two points on a circle, is a chord of the circle. Two or more circles with the same center are concentric circles.

An

angle formed by two radii of a circle is called a central angle. A central angle opens to a portion of a circle called an arc, which is part of the circumference of a circle. The region enclosed by an arc and its central angle is called a sector.

An

on the circumference of a circle and whose sides are chords of the circle is an inscribed angle. A polygon is inscribed in a circle if all of its vertices are on the angle

whose vertex

circumference of the

is

circle.

Lesson 21

1

37

LESSON

21

Prime and Composite Numbers Prime Factorization

WARM-UP Facts Practice: Circles (Test E)

Mental Math: a.

$1.25 + 99C

d.

6 x 34

g.

10

$6.50

b.

c.

$20.00 - $15.75

f. if + 2| f of 36 Start with the number of sides of a hexagon, x 5, +

+

1,

7

e.

2,

8,

-r

5,

Problem Solving:

Sam can read 20 pages in take Sam to read 200 pages? If

30 minutes,

how many

hours will

it

NEW CONCEPTS Prime and composite

numbers

We remember that the counting numbers are the

numbers we use

They

to count.

(or natural

numbers)

are

1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

...

Counting numbers greater than 1 are either prime numbers or composite numbers. A prime number has exactly two different factors, and a composite number has three or more factors. In the following table

we

list

the factors of the

first

ten counting numbers. The numbers 2, 3, 5, and 7 each have exactly two factors, so they are prime numbers. Factors of Counting

Number

Numbers 1-10 Factors

1

1

2

1.2

3

1,

4

1,2,4

5

1,5

6

1, 2, 3,

7

1,7

8

1, 2, 4,

9

1, 3,

10

3

6

8

9

1,2, 5, 10

1

38

Saxon Math 8/7

We see that the the

number

f actors

itself.

A

numbers are 1 and prime number as follows:

of each of the prime

So we define a

prime number

is

whose only number itself.

greater than 1

and the

From

the table

number

a counting

factors are 1

we can

also see that 4, 6, 8, 9, and 10 each factors, so they are composite numbers.

have three or more Each composite number and itself.

is

divisible

by a number other than

Note: Because the number 1 has only one factor, itself, neither a prime number nor a composite number.

Example

1

Solution

Make First

a

list

we

of the prime

list

numbers

the counting

it is

that are less than 16.

numbers from

1, 2, 3, 4, 5, 6, 7, 8, 9,

1

1 to 15.

10, 11, 12, 13, 14, 15

A

prime number must be greater than 1, so we cross out 1. The next number, 2, has only two divisors (factors), so 2 is a prime number. However, all the even numbers greater than 2 are divisible by 2, so they are not prime. We cross these out.

n,

t, 2, 3, 4, 5, 0, 7, 0, 9, Kf, 11,

The numbers

H,

15

that are left are 2, 3, 5, 7, 9, 11, 13,

The numbers

13,

9

and 15

are divisible

by

15

3,

2, 3, 5, 7, 0, 11, 13,

so

we

cross

them

out.

V*

The only divisors of each remaining number are 1 and the number itself. So the prime numbers less than 16 are 2, 3, 5, 7, 11, and 13. Example

2

Solution

List the First

we

composite numbers between 40 and write the counting

50.

numbers between 40 and

50.

41, 42, 43, 44, 45, 46, 47, 48, 49

by a number besides 1 and itself is composite. All the even numbers in this list are composite since they are divisible by 2. That leaves the odd

Any number

that

is

divisible

numbers to consider. We quickly see that 45 is divisible by So both 45 and 49 are 5, and 49 is divisible by 7.

Lesson

21

1

39

composite. The remaining numbers, 41, 43, and 47, are prime. So the composite numbers between 40 and 50 are

and

42, 44, 45, 46, 48,

Prime factorization

49.

Every composite number can be composed (formed) by multiplying two or more prime numbers. Here we show each of the first nine composite numbers written as a product of prime number factors. 4 =

2-

2-

6 =

2

9=3-3 14 =2-7

10 15

8 =

3

=2-5 =3-5

Notice that we factor 8 as 2 4 is not prime.

•

16 •

2

2

=2-2-3 =2-2-2-2

12

2

2- 2-

and not 2-4, because

When we write a composite number as a product of prime numbers, we are writing the prime factorization of the number. Example

3

Write the prime factorization of each number. (a)

Solution

We

30

(b)

81

number

will write each

420

(c)

two or more

as the product of

prime numbers. (a)

30 = 2

3

•

We

6 or 3 do not use 5 because neither 6 nor 10

5

•

•

•

10, is

prime. (b)

81 = 3

3

•

3

•

We do not use

3

•

is (c)

420 = 2

•

2

•

3

•

5

•

7

Solution

9,

•

because 9

not prime.

Two methods are

Example 4

9

shown

after

for finding this

example

Write the prime factorization of 100 and of Vl00

4.

.

We

find 5. 2 5 factorization of 100 is 2 5. that a 100 is 10, and the prime factorization of 10 is 2 Notice that 100 and VI 00 have the same prime factors, 2 and 5, but that each factor appears half as often in the prime

The prime

•

•

•

•

factorization of vlOO.

There are two commonly used methods for factoring composite numbers. One method uses a factor tree. The other method uses division by primes. We will factor 420 using both methods.

1

40

Saxon Math 8/7

number using a factor tree, we first write the number. Below the number we write any two whole numbers To

factor a

greater than 1 that multiply to equal the

number.

If

these

numbers are not prime, we continue the process until there is a prime number at the end of each "branch" of the factor tree. These numbers aie the prime factors of the original number. We write them in order from least to greatest. Factor Tree

420

420 = 2

•

2

3

•

•

5

•

7

To factor a number using division by primes, we write the number in a division box and divide by the smallest prime number that is a factor. Then we divide the resulting quotient by the smallest prime number that is a factor. We repeat this + process until the quotient is l. The divisors are the prime factors of the

number. Division by Primes 1

7J7 5j35 3ll05 2)210 2)420

420 = 2

•

2

•

3

•

5

•

7

LESSON PRACTICE Practice set*

a.

List the first ten

b. If a

whole number

kind of number

c. +

prime numbers. greater than 1

is

not prime, then what

is it?

Write the prime factorization of 81 using a factor

Some people

prefer to divide until the quotient

the fina quotient 1

is

included in the

list

is

a prime

of prime factors.

tree.

number. In

this case,

Lesson 21

1

41

d.

Write the prime factorization of 360 using division by primes.

e.

Write the prime factorization of 64 and of V64.

MIXED PRACTICE Problem set

1. (14)

Two

thirds of the students

Day.

What

fraction

St. Patrick's

2. {13]

3. l5,

wore green on St. Patrick's of the students did not wear green on

Day?

Three hundred forty-three quills were carefully placed into 7 compartments. If each compartment held the same number of quills, how many quills were in each compartment?

How much less than

121

write the answer.

4.

Last year the price

(n)

5.

increased $824.

2 billion is 21 million?

Use words

was $14,289. This year the

What

is

to

price

the price this year?

Write each number as a reduced fraction or mixed number:

"5 '

I

(a)

6. List

3

(b)

|f

the prime

12%

(c)

if

numbers between 50 and

60.

(21)

7.

Write the prime factorization of each number:

(21)

(a)

8. 141

50

(b)

Which

How

60

number

point could represent 1610 on this did you decide?

ABC

D 2000

1000

9.

300

(c)

Complete each equivalent

fraction:

(15)

(a)

10. (a)

I = T5

(b)

I =

How many ^'s

are in 1?

How many ^'s

are in 3?

(9)

(b)

A

(C)

I = T2

line?

142

Saxon Math 8/7

11. 1201

12. 8 19) '

13.

The perimeter

What Use

is its

twice as long as

How

(b)

What

long is

is

draw it is

a rectangle that

is

f in.

wide and

wide.

the rectangle?

the perimeter of the rectangle?

Find the perimeter of this hexagon:

5

8

3

Draw

in.

in.

12

14.

12 inches.

is

area?

a ruler to

(a)

of a regular quadrilateral

in.

in.

a pentagon.

(18)

Solve: 15. (9)

18 (9,

15)

p + - = F 5

1

16. (9)

i + / = | 6 6

19.

|q =

17.

1

5

272

ftj

- 3| = if

i

20. 51

= 50

= 3c

Simplify: 21. (g)

23.

+ § + | | 3 3 3

22.

f

(a)

Write the prime factorization of 225.

(b)

Find a/225 and write

(21)

prime

factorization.

finding the greatest common factor of the numerator and denominator of a fraction can help reduce

24. Describe 1151

its

how

the fraction.

25. ll7)

2§ inches long. Then draw BC l\ inches long perpendicular to AB. Complete the triangle by drawing

Draw AB

AC. Use a protractor

to find the

measure of ZA.

Lesson 21

1

43

26. Write if as an improper fraction. Multiply the improper fraction by the reciprocal of |. Then write the product as

(9 101 '

a

mixed number.

27. Refer to the circle at right

'

center at point

Mto answer

(a)

Which segment

(b)

Which segment

with

(a)-(d).

a diameter?

is is

a chord but

not a diameter?

28.

(c)

Which two segments

(d)

Which

angle

is

are radii?

an inscribed angle?

A quart is what percent of a gallon?

(16)

29.

(a)

Compare: a +

bQ b

(b)

What property

of operations applies to part

+ a

(2)

(a)

of this

problem? 30. Refer to the triangles

below

to

answer

(a)-(c).

(18)

(a)

Which

triangle appears to be congruent to

(b)

Which

triangle

is

(c)

Which

angle in

AQRS corresponds

not similar to

AABC?

AABC? to

ZA in AABC?

1

44

Saxon Math 8/7

LESSON

22

Problems About a Fraction of a Group

WARM-UP Facts Practice: Circles (Test E)

Mental Math: 100

$10.00 - $7.89

a.

$1.54 + 99

Room

there are 28 students. In Room 9 there are 30 students. In Room 11 there are 23 students. How many students are in all three rooms? In

the students in problem 1 were equally divided

2. If all

m

three rooms,

3.

One hundred

4. {5

'

7

12>

5. (22}

how many students would be

in each

among

room?

twenty-six thousand scurried through the colony before the edentate attacked. Afterward only seventy-nine thousand remained. How many were lost when the edentate attacked?

Two

thousand, seven hundred is how much less than ten thousand, three hundred thirteen? Use words to write the answer.

Diagram

this

statement.

Then answer

the questions

that follow.

Five ninths of the 36 spectators were

happy

with

the outcome. (a)

(b)

How many spectators were happy with the How many spectators were not happy

outcome? with the

outcome? 6. {22)

In

change the percent to a Then diagram the statement and answer

the following statement,

reduced

fraction.

the questions.

Twenty-five percent of the three dozen eggs were cracked. (a)

What

(b)

How many

fraction of the eggs

were not cracked?

eggs were not cracked?

'

1

48

Saxon Math 8/7

7.

(a)

What

(15)

is

(b)

fraction of the rectangle

shaded?

What percent

of the rectangle

is

not shaded? 8.

(a)

How many ^'s

(b)

Use the answer to

(a)

Multiply:

(b)

What property

are in 1?

(9)

9.

part

to find the

(a)

number

of ^'s in

3.

6-5-4-3-2-1-0

(2)

is

illustrated

by the multiplication

in

part (a)?

10. Simplify

11.

o

O

(9)

(19, 20)

3^

.

^OJ

-r

|,

3

5

8

4

9

15

8 ^ 6 5

5

iM% »

instead of dividing 5 by f

find the answer by multiplying 5 by

,

we can

what number?

1

68

Saxon Math 8/7

27.

Simplify and compare:

(a)

(20)

ll9)

2

3

:

2

.

(b)

28.

2'

Simplify: V2

2

A

regular hexagon is inscribed in a circle. If one side of the hexagon is 6 inches long, then the perimeter of

the hexagon

29. ll9)

A

2-in.

square

What

is

how many

feet?

square was cut from a 4-in. as

shown

in

the

perimeter resulting polygon? is

the

4

figure.

of

2

the

2

30. {4

-

21)

Which

negative integer

number?

is

in.

in.

in.

the opposite of the third prime

Lesson 26

1

69

LESSON

26

Multiplying and Dividing

Mixed Numbers WARM-UP Facts Practice: Lines, Angles, Polygons (Test F)

Mental Math: a.

$8.56 +

d.

3 x 74

g.

7 x

98

q w

are

16.

12

360°

4 i

^

whole numbers?

180 - y = 75

(3)

18.

w

+ 58- = 100 3

(23)

(3)

19. (a)

10

Find the area of the square.

(20)

(b)

Find the area of the shaded part of the square.

Simplify: 20. (23)

4 1 9^ - 4^

22. (2|) (20. 26)

9

9

2

How

least to greatest:

Solve: 15.

long

AC?

10)

(b)

in.

21 r«)

5 8

.

S

23.

A 10

.

1 6

2

if * 2|

in.

1

74

Saxon Math 8/7

24. 3^ (26)

26.

25. 5

4

-r

3

V10

2 •

10

16,524

4

27.

36

(i)

(20)

28. Evaluate the following expressions for

x -

3

and y =

6:

(1.9)

(b)

(cj

(2)

The

rule of the following sequence

the ninth term. 1,4,

30. tinv.2)

The 18Q

o

7,

10,

...

central angle of a half circle is

The centra i ang } e

circle is 90°.

f a

q Uar ter

How many

degrees is the central angle of an eighth of a circle?

•

7

7 29.

- -

1

is

^

k = 3n -

2.

Find

Lesson 27

1

75

LESSON Multiples • Least Common Multiple • Equivalent Division Problems WARM-UP Facts Practice: Circles (Test E)

Mental Math: a.

S3. 75 + SI. 98

d.

5

g.

Start

x

b.

S125.00 + 10

10

c.

42

f.

x

42

| of 24

with a score. Add a dozen: then add the number of feet in a yard. Divide by half the number of years in a decade; then subtract the number of days in a week. What is the answer?

Problem Solving:

Simon held a faces. Simon have been

on three adjoining dots. Could Simon

die so that he could see the dots

said he could see a total of 8

telling the truth?

Why

or

why

not?

NEW CONCEPTS Multiples

The multiples of a number are produced by multiplying the number by 1. by 2, by 3, by 4, and so on. Thus the multiples of 4 are 4. 8, 12, 16, 20. 24, 28, 32, 36,

The multiples

...

of 6 are 6. 12. 18. 24. 30. 36.

42. 48. 54,

...

inspect these two lists, we see that some of the numbers in both lists are the same. A number appearing in both of these lists is a common multiple of 4 and 6. Below we have circled some of the common multiples of 4 and 6. If

we

We

@, 16. 20, @, 28. 32. @, @, 18. @, 30, (36). 42. 48. 54,

Multiples of

4:

4. 8,

Multiples of

6:

6,

see that 12. 24. and 36 are

we continued both common multiples. If

...

lists,

...

common multiples of 4 and 6. we would find many more

1

76

Saxon Math 8/7

Least

common multiple

the least (smallest) of the common multiples. The least common multiple of 4 and 6 is 12. Twelve is the smallest number that is a multiple of both 4 and

Of

6.

Example

1

Solution

particular interest

The term

Find the

We

will

list

common

least

least

is

common

multiple

is

often abbreviated

multiple of 6 and

some multiples

8.

and of 8 and

of 6

LCM.

common

circle

multiples. 6:

6, 12, 18,

Multiples of

8:

8, 16, (24), 32,

We find that the It is

Solution

least

unnecessary

for the least

Example 2

@,

Multiples of

Find the

common

@,

...

56, 64,

...

30, 36, 42,

40,

@,

multiple of 6 and 8

is

24.

each time. Often the search multiple can be conducted mentally.

to list multiples

common

LCM of 3,

4,

and

6.

common

multiple of 3, 4, and 6, we can mentally search for the smallest number divisible by 3, 4, and 6. We can conduct the search by first thinking of multiples of

To

find the least

the largest number,

6.

6, 12, 18, 24,

Then we mentally

...

multiples for divisibility by 3 and by 4. We find that 6 is divisible by 3 but not by 4, while 12 is divisible by both 3 and 4. Thus the LCM of 3, 4, and 6 is 12. test these

We

can use prime factorization to help us find the least common multiple of a set of numbers. The LCM of a set of numbers is the product of all the prime factors necessary to

form any number in the

Example 3 Solution

Use prime

set.

factorization to help

We write the prime 18

factorization of 18

=2-3-3

LCM of 18 and 24.

you find the

24

and of

24.

=2-2-2-3

The prime factors of 18 and 24 are 2's and 3's. From a pool of three 2's and two 3's, we can form either 18 or 24. So the LCM of 18 and 24 is the product of three 2's and two 3's.

LCM of 18

and 24 =

2

= 72

•

2

-

2

•

3

•

3

Lesson 27

Equivalent

Tricia's teacher

division

T£ If

asked this question: „ ,

.

.

.

,

quickly gave the correct answer, explained how she found the answer.

Tricia

77

what

sixteen flavored icicles cost S4.00, was the price for each flavored icicle?

problems

1

25c,

and then

knew I had to divide S4.00 by 16. but I did not know the answer. So I mentally found half of each number, which made the I

problem S2.00

couldn't think of the answer, so I found half of each of those numbers. That made the problem $1.00 4, and I knew the answer was 250. 8. 1 still

-f-

How

did Tricia's mental technique work? Recall from Lesson 15 that we can form equivalent fractions by multiplying or dividing a fraction by a fraction equal to 1.

3

10 10

X

4

30

40

6

9

3

2

"

3

3

We can form equivalent division problems in a similar way. We multiply (or divide) the dividend and divisor by the same form a

new

calculate mentally.

The

number the

same

to

quotient, as

$4.00

H-

16 t 2

Example 4

Solution

Example

5

Solution

2

division problem that is easier to new division problem will produce

we show below.

$2.00 8

$2.00 - 2 8-1-2

$1.00 4

= 50,25

Instead of dividing 220 by 5, double both numbers and mentally calculate the quotient.

We We

10. double the two numbers in 220 -f 5 and get 440 mentally calculate the new quotient to be 44, which is also the quotient of the original problem.

Instead of dividing 6000 by 200, divide both numbers by 100, and then mentally calculate the quotient.

We

mentally divide by 100 by removing two places (two zeros) from each number. This forms the equivalent division 2. We mentally calculate the quotient as 30. problem 60

1

78

Saxon Math 8/7

LESSON PRACTICE Practice set

common

Find the least of numbers: a.

8

multiple (LCM) of each pair or group

and 10

Use prime

b. 4, 6,

and 10

LCM

you find the

factorization to help

of these

pairs of numbers:

30 and 75

c.

24 and 40

e.

Instead of dividing 7| by l|, double each mentally calculate the quotient.

d.

number and

Mentally calculate each quotient by finding an equivalent division problem. Discuss your strategy with the class. f.

24,000

400

-r

$6.00

g.

12

4-

h.

140

-f

5

MIXED PRACTICE Problem set

1. 1111

2.

There were three towns in the valley. The population of Brenton was 11,460. The population of Elton was 9420. The population of Jennings was 8916. What was the total population of the three towns in the valley?

Norman

6 feet

is

tall.

How many inches tall

is

Norman?

(13, 16)

the cost of one dozen eggs was $1.80, what was the cost per egg? Write an equivalent division problem that is

3. If t27)

easier to calculate mentally, 4. (5,

Which

and find the

quotient.

of the following equals one billion?

20)

A. 10 5. {22)

3

B. 10

Diagram

this

6

statement.

C. 10

9

Then answer

D. 10

12

the questions

that follow.

Three eighths of the 712 students bought their lunch. (a)

(b)

6. (19 20) -

How many students bought their lunch? How many students did not buy their lunch?

The perimeter

of this rectangle

is

30 inches. (a)

What

6

is

the

length

of

the

rectangle? (

j)

What

is

the area of the rectangle?

in.

179

Lesson 27

7. (27)

8.

Use

prime

factorization

find

to

the

least

common

multiple of 25 and 45.

What number

is

halfway between 3000 and 4000?

(4)

9.

24%

reduced

(a)

Write

(b)

Use prime factorization

as a

fraction.

(15. 24)

10. 1161

was

It

a "scorcher."

to

reduce

36 180

The temperature was 102°F

in the

shade. (a)

The temperature was how many degrees above the freezing point of water?

(b)

The temperature was how many degrees below the boiling point of water?

each fraction, write an equivalent fraction that has a denominator of 36.

11. For (2 15) '

Wi (d)

(b)

(c)

§

What property do we use when we

i find equivalent

fractions? 12. (a) Write the

prime factorization of 576 using exponents.

(21)

(b)

Find

a

and 6|

13. Write 5§ (26)

576 as

improper fractions and find

their

product.

In the

below, quadrilaterals

figure

squares. Refer to the figure to

14. (a) (7)

(b)

15. (a) (b) (c)

16. If

E

F

A

D

C

B

ZACD?

of angle

Name two

segments parallel

AB is

9 20)

answer problems 14-16.

What kind

What What What 3

ABCF and FCDE

is

fraction of square

to FC.

CDEF is shaded?

ABCF is shaded? of rectangle ABDE is shaded?

fraction of square fraction ft,

ABDE? ABDE?

(a)

what

is

the perimeter of rectangle

(b)

what

is

the area of rectangle

'

are

180

Saxon Math 8/7

Solve: 17.

lOy = 360°

(3,

(3)

19.

5^ - n = if

2

4

=

12'

20)

20.

O

O

(23)

p +

18.

277

- 6- =

4 3

(10)

Simplify: 21. 10

5

23.

(26)

a2

24.

3

22.

-I 6

25.

9

5

6

8

'

(26)

4 15

8

26. f9.

7

5

f

9

15)

9

27. If the diameter of a circle is half of a yard, then its radius tinv.2)

is

now man y

i

ncnes ?

28. Divide $12.00 {27]

29. (i9, 20)

by 16 or find the quotient of an equivalent

division problem.

A 3-by-3-in. paper square is cut from a 5_ D y_5_i n paper square as shown. (a)

What

is

3 5

the perimeter of the

in.

3

in.

in.

resulting polygon? (b)

How many

square inches of the 5-by-5-in. square remain?

30. Refer to this circle

with center

at

point

5

in.

M to answer

(a)-(e):

(Inv. 2)

(a)

Which chord

is

a diameter?

(b)

Which chord

is

not a diameter?

(c)

What

(d)

Which

(c)

Which two

angle

is

an acute central angle?

angles are inscribed angles? sides of triangle

AMB are equal in length?

181

Lesson 28

LESSON

28

Two-Step Word Problems Average, Part 1

WARM-UP Facts Practice: Lines, Angles. Polygons (Test F)

Mental Math: a.

S6.23 + S2.99

b.

SI. 75 x

d.

8 x 53

e.

| 8

g.

Think of an

+ |8

100

c. f.

S5.00 - SI. 29 4 of 25 o

easier equivalent division for S56.00

-r

14.

Then

find the quotient.

Problem Solving:

When

Bill. Phil. Jill,

and

Gil entered the room, they

found four

They each took a seat, and one minute Then they traded seats again and again. If

chairs waiting for them.

they traded seats. they don't move the chairs but only move themselves, seating arrangements (permutations) are possible? later

how many

MEW CONCEPTS Two-step

word

Thus

far

we have

considered six one-step word-problem

themes:

problems 1.

Combining

2.

Separating

3.

Comparing

4.

Elapsed Time

5.

Equal Groups

6.

Part of a

Whole

Word problems often require more than one step to solve. In this lesson we will continue practicing problems that require multiple steps to solve. These problems involve two or more of the themes mentioned above.

1

82

Saxon Math 8/7

Example

1

Julie

went

food for Solution

This

is

to the store

67(2

how much

per can,

she bought 8 cans of dog money did she have left?

with $20.

a two-step problem. First

spent. This first step Number Number

is

of

we

find out

how much Julie

an "equal groups" problem.

groups

—— ——

Total

—^

in

If

group

$0.67 each can X 8 Cans

$5.36

can find out how much money Julie had second step is about separating.

Now we

left.

This

$20.00 - $5.36 $14.64 After spending $5.36 of her $20 on dog food, Julie had

$14.64

Average, part

1

left.

Calculating an average

is

often a two-step process.

As an

example, consider these five stacks of coins:

There are 15 coins in all. If we made all the stacks the same size, there would be 3 coins in each stack.

We

say the average number of coins in each stack look at the following problem:

is 3.

Now

There are 4 squads in the physical education class. Squad A has 7 players, squad B has 9 players, squad C has 6 players, and squad D has 10 players. What is the average number of players per squad?

The average number of players per squad is the number of players that would be on each squad if all of the squads had

Lesson 28

1

83

same number of players. To find the average of a group of numbers, we begin by finding the sum of the numbers. the

7 players

9 players 6 players

+ 10 players 32 players

Then we divide

numbers. There are

sum

sum

numbers by the number 4 squads, so we divide by 4.

the

of the

of

numbers _ 32 players number of numbers 4 squads of

= 8 players per squad Finding the average took two steps. First we added the numbers to find the total. Then we divided the total to make equal groups.

Example

2

Solution

When

people were seated, there were 3 in the first row, 7 in the second row, and 20 in the third row. What was the average number of people in each of the first three rows?

The average number of people in the first three rows is the number of people that would be in each row if the numbers were equal. First we add to find the total number of people. people 7 people + 20 people 3

30 people

Then we divide by

3 to separate the total into 3 equal groups.

30 people _ 1Q e0 pi e er row p p 3

rows

The average was 10 people

in each of the

that the average of a set of

smallest the set.

number in

numbers

rows. Notice greater than the

first 3

is

the set but less than the largest

number in

Another name for the average is the mean. We find the mean of a set of numbers by adding the numbers and then dividing the sum by the number of numbers.

1

84

Saxon Math 8/7

Example

3

On

the last test five students in the class scored 100, four scored 95, six scored 90, and five scored 80. What was the

mean Solution

First

of the scores?

we

find the total of the scores.

100 = 4 x 95 = 6 x 90 = 5 x 80 =

500 380 540 400

5 x

1820

Next in

we

divide the total by 20 because there were 20 scores

all.

sum

numbers number of numbers

We

find that the

1820 _ 20

of

mean

of the scores

was

91.

LESSON PRACTICE Practice set

Work each problem

as a two-step problem:

with $20 and returned home with $5.36. If all she bought was 3 bags of dog food, how much did she pay for each bag?

a.

Jody went

b.

Three eighths of the 32 students in Mr. Scaia's class were girls. How many boys were in Mr. Scaia's class?

c.

In

Room

to the store

1

there were 28 students, in

were there were 30 students, and in

29 students, in Room 3 Room 4 there were 25 students. number of students per room? d.

What

e.

What

is

is

the

Room

mean

2 there

What was

of 46, 37, 34, 31, 29,

the average of 40 and 70?

the average

and 24?

What number

is

halfway between 40 and 70? f.

his highest score

was

95.

his average test score?

A. 80

B.

84

was

and Which of the following could be

Willis has taken eight tests. His lowest score

Why? C. 95

D. 96

80,

Lesson 28

1

85

MIXED PRACTICE Problem set

1.

m

The

weighed 242 pounds, 236 pounds, 248 pounds, 268 pounds, and 226 pounds. What was the average weight of the players on the front on the front

5 players

line

line? 2. (28}

Yuko ran

How many

minutes 14 seconds. take Yuko to run a mile?

a mile in 5

seconds did

it

3.

Luisa bought a pair of pants for $24.95 and 3 blouses for $15.99 each. Altogether, how much did she spend?

4.

Columbus was 41 years old when he reached the Americas in 1492. In what year was he born?

m {u]

5. (22)

The

In

Italian navigator Christopher

change the percent to a Then diagram the statement and answer

the following statement,

reduced

fraction.

the questions.

Salma led for 75% of the 5000-meter race.

6. (19 20) '

how many

(a)

Salma led the race

(b)

Salma did not lead the race

This rectangle is wide. (a)

(b)

What

is

for

how many

for

twice as long as

meters?

it

the perimeter of the rectangle? is

What

is

the

^

'

the

of

area

meters?

rectangle? 7.

(a)

List the first six multiples of 3.

(27)

(b) List

the

first

(c)

What

(d)

Use prime

is

the

six multiples of 4.

LCM

of 3

and 4?

factorization to find the least

multiple of 27 and 36. 8.

On (a)

number line below, 283 which multiple of 10?

(b)

which multiple

(27)

the

is

closest to

of 100? 283

—— — — — — — — — — —h»H— —h— I

I

200

I

I

I

I

I

I

I

I

300

common

1

86

Saxon Math 8/7

9. l24)

10. (16)

Write 56 and 240 as products of prime numbers. Then reduce J|.

A

mile is five thousand, two hundred eighty feet. Three feet equals a yard. So a mile is how many yards?

(15)

(a)

(a)

12.

that has a

and (b), find an equivalent fraction denominator of 24.

11. For

(b)

I

g

I

(c)

Add

(a)

Write the prime factorization of 3600 using exponents.

(b)

Find V3600.

you found.

the two fractions

(21)

13. Describe

how to

find the

mean

of 45, 36, 42, 29, 16,

and

24.

(28)

14. (8,

Draw square ABCD so that each What is the area of the square?

(a)

20)

Draw segments

(b)

AC and

side

inch long.

is 1

BD. Label the point

at

which

they intersect point E. (c)

Shade

(d)

What percent

triangle

CDE.

Arrange these numbers in order from

15. (a)

you shade?

of the area of the square did

least to greatest:

(4,10)

4±

Which

(b)

of these

n u

_L '

i 11 10' A ' 10'

numbers

are

odd

integers?

Solve: 16.

12y = 360°

17. 10

2

=

m

+ 8

2

18.

(3,20)

(3)

(3)

—W

Simplify: 19. (9,15)

4^- -

21. 12

20.

i-£

(10,15)

8

- 8^

22.

6?

8

(26)

3

(23)

\

8

•

15

2

23. \l\ ^ V (20, 26)

1 - 77-

24.

8^2-2

2

(26)

3

26

25. (D

8- + 3-

1-^-

±£

80

.

(25)

1 4

^2

3

= 60

1

87

and y =

4:

Lesson 28

27. Evaluate the following expressions for (b)

(a)

28.

Draw

2

x =

x + y

3

2

a decagon.

29. In the figure

below the two

triangles are congruent.

(18)

A

B

C

D (a)

(b) (c)

AACD corresponds to ZCAB in AABC? Which segment in AABC corresponds to AD in AACD? 2 If the area of AABC is 7| in. what is the area of figure

Which

angle in

,

ABCD? 30. (l 7)

With a ruler draw PQ 2| in. long. Then with a protractor draw QJR so that ZPQB measures 30°. Then, from point P, draw a ray perpendicular to PQ that intersects QB. (You may need to extend QB to show the intersection.) Label the point where the rays intersect point M. Use a protractor to measure ZPMQ.

1

88

Saxon Math 8/7

LESSON

29

Rounding Whole Numbers Rounding Mixed Numbers Estimating Answers

WARM-UP Facts Practice: Circles (Test E)

Mental Math: a.

$4.32 + $2.98

b.

d.

9 x 22

e.

g.

6

x

6,

-r

4,

X

3,

+

10

$12.50

c.

+ I6 1,

-r

4,

x 8,

-

1,

$10.00 - $8.98

f.

I of

-r

5,

20

X

2,

-

2,

-r

2

Problem Solving:

Huck followed

the directions on the treasure map. Starting at

left, and walked and walked five paces, turned left again, and walked four more paces. He then turned right and took one pace. In which direction was Huck facing, and how many paces was he from the big tree?

the big tree, he walked six paces north, turned

seven more paces. He turned

left

NEW CONCEPTS Rounding whole numbers

The

sentence below uses an exact number to state the size of a crowd. The second sentence uses a round number. first

There were 3947 fans at the game. There were about 4000 fans at the game.

Round numbers are often used instead of exact numbers. One way to round numbers is to consider where the number is located on the number line. Example

1

Solution

Use

a

number

line to

(a)

round 283

to the nearest

(b)

round 283

to the nearest ten.

(a)

We

hundred.

draw a number line showing multiples mark the estimated location of 283.

of 100 and

283 100

We

200

300

400

between 200 and 300 and is closer 300. To the nearest hundred, 283 rounds to 300. see that 283

is

to

Lesson 29

(b)

We

draw a number line showing the tens from 200 and mark the estimated location of 283.

1

to

89

300

283 -«

We

1

1

1

1

1

200

210

220

230

240

1

1

1

250

260

270

f-4

280

1

1

290

300

between 280 and 290 and 280. To the nearest ten, 283 rounds to 280. see that 283

Sometimes we

is

are asked to

round

a

number

is

closer to

to a certain place

We can use an underline and a circle to help us do We will underline the digit in the place to which we are

value. this.

rounding, and we will circle the next place to the we will follow these rules: 1.

If

the circled digit

is

or more,

5

underlined digit. If the circled digit leave the underlined digit unchanged. 2.

We

replace the circled digit and

right.

we add is

less

all digits to

1

than

Then

to 5,

the

we

the right of

the circled digit with zeros.

This rounding strategy is sometimes called the "4-5 split," because if the circled digit is 4 or less we round down, and if it is

Example

2

Solution

5 or

more we round up.

(a)

Round 283

to the nearest

(b)

Round 283

to

(a)

We underline the we

hundred.

the nearest ten. 2 since

it is

in the

hundreds place. Then

circle the digit to its right.

2®3 Since the circled digit is 5 or more, we add 1 to the underlined digit, changing it from 2 to 3. Then we replace the circled digit and all digits to its right with zeros and get

300 (b)

Since

we

are rounding to the nearest ten,

the tens digit and circle the digit to

we

underline

its right.

28(3)

than 5, we leave the 8 replace the 3 with a zero and get

Since the circled digit

unchanged. Then

we

is

less

280

1

90

Saxon Math 8/7

Example

3

Solution

Example 4 Solution

Rounding mixed numbers

Round 5280

so that there

Round 93,167,000 To the nearest

answers

million, 93,(1)67,000 rounds to 93,000,000.

whole number, we need part of the mixed number a

determine whether the fraction is greater than, equal to, or less than |. If the fraction is greater than or equal to |, the mixed number rounds up to the next whole number. If the fraction is less than |, the mixed number rounds down. to

greater than \ if the numerator of the fraction fraction is less than \ than half of the denominator.

fraction

is

A

Round 14^

is

less

The mixed number 14^ is between the consecutive whole numbers 14 and 15. We study the fraction to decide which is nearer. The fraction ^ is greater than \ because 7 is more than

14^ rounds

to 15.

Rounding can help us estimate the answers to arithmetic problems. Estimating is a quick and easy way to get close to an exact answer. Sometimes a close answer is "good enough," but even when an exact answer is necessary, estimating can help us determine whether our exact answer is reasonable. One way to estimate is to round the numbers before

Mentally estimate: (a)

Solution

if

whole number.

calculating.

Example 6

is

than half of the denominator.

to the nearest

half of 12. So

Estimating

to the nearest million.

When rounding a mixed number to

the numerator

Solution

digit.

round the number so that all but one of the digits are zeros. In this case we round to the nearest thousand, so 5280 rounds to 5000.

more

5

one nonzero

We

A

Example

is

(a)

5^

x 3|

We

(b)

396 x 312

round each mixed number to the nearest whole number before

we

4160

(c)

5 7

x

-f

19

3I

10

3

multiply. J

|

6 x 3 = 18

(b)

Lesson 29

191

can round each number to the nearest ten or nearest hundred.

to the

We

Rounded

Problem 396 x 312

x

Rounded

to tens

400 310

x

to

hundreds

400 300

When mentally

estimating we often round the numbers to one nonzero digit so that the calculation is easier to perform. In this case we round to the nearest hundred.

x

400 300

120,000 (c)

We round each number so there is we

one nonzero

digit before

divide.

4160

4000

19

20

= 200

Performing a quick mental estimate helps us determine whether the result of a more complicated calculation is reasonable.

LESSON PRACTICE Practice set

a.

Round 1760

to the nearest

b.

Round 5489

to the nearest thousand.

c.

Round 186,282

hundred.

to the nearest thousand.

Estimate each answer: d.

f.

7986 - 3074

5860

t"

e.

297 x 31

19

h. Calculate the area of this rectangle.

the check reasonableness of your answer by using round numbers to estimate

After

the area.

calculating,

l|in.

1

1

92

Saxon Math 8/7

MIXED PRACTICE Problem set

1. (16,28)

2. (13)

3. (28>

4. (5

'

121

Lorenz jumped 16 feet 8 inches on his man y inches did he jump on his first try?

pounds of bananas cost per pound? If

8

cost $3.68,

how

first try.

we

can

How

find the

On her first six tests

Sandra's scores were 75, 70, 80, 80, 85, and 90. Find the mean of these six scores.

Two hundred nineteen billion, eight hundred million is how much less than one trillion? Use words to write your answer.

5. {22]

following statement, change the percent to a reduced fraction. Then diagram the statement and answer the questions. In the

Forty percent of the 80 chips were blue.

6.

(a)

How many of the

chips were blue?

(b)

How many of the

chips were not blue?

(a)

What

{27]

is

the least

common

multiple (LCM) of

4, 6,

and 8? (b)

Use prime

LCM

factorization to find the

of 16 and

36.

7.

(a)

What

is

the perimeter of this

square? (b)

What

is

the area of this square? 1

1

1

1

1

1

1

1

1

1

1 1

1 1

inch

8.

(a)

Round 366

(b)

Round

to the nearest

""I

hundred.

(29)

9. t29)

36ji to the nearest ten.

Mentally estimate the sum of 6143 and 4952 by rounding each number to the nearest thousand before adding.

Lesson 29

10. (a) (26, 29)

1

93

Mentally estimate the following product by rounding each number to the nearest whole number before multiplying:

1

.

5i

4 (b)

Now

ll

.

8

3

find the exact product of these fractions and

mixed numbers. 11.

Complete each equivalent

fraction:

(15)

2 =

(a)

3

12. (2o,2i)

± 30

The prime p r j me

M

1 = 6

factorization of 1000

25 30 3

3

Write the factorization of one billion using exponents. is

2

•

5

.

In the figure below, quadrilaterals

ACDF, ABEF, and BCDE

are rectangles. Refer to the figure to

answer problems 13-15.

A

B

C

F

E

D

What percent

of rectangle

ABEF is

shaded?

(b)

What percent

of rectangle

BCDE is

shaded?

(c)

What percent

of rectangle

A CDF is

shaded?

13. (a) (8)

14. 9,20)

relationships between the lengths of the sides of the rectangles are as follows:

The

AB >+ FE

=

BC

AF

=

AC

+

AB

CD =

2 in.

(a)

Find the perimeter of rectangle ABEF.

(b)

Find the area of rectangle BCDE.

15. Triangle

ABF is

congruent to AEFB.

' 18>

(a)

Which

(b)

What

angle in

is

AABF corresponds

the measure of

ZA?

to

ZEBF in AEFB?

Saxon Math 8/7

Solve: 16. 8

2

=

x + 4^ = 15

17.

4217

(23)

(3,20)

18.

3^ = n - 4-

9

(10, 15)

9

5

21.

l|

(26)

O

9

Simplify: 19.

6± - 5§ O

(23)

20.

O

6§

-f

O

(26)

-f

3± Z

22. $7.49 x 24 23. Describe 24.

how to

estimate the product of 5| and 4|.

Find the missing exponents.

(20)

10

(a)

3 •

10

3

g

= 10 m

(b)

^10

25. {2]

The

rule of the following sequence

how

you inscribed hexagon in a circle

26. Recall inv. 2)

is

k =

2

n

+

Find

1.

the fifth term of the sequence. 3, 5, 9, 17,

do,

= 10 n

3

re g U j ar

Investigation

2.

If

...

a

in

the radius of

this circle is 1 inch, (a)

what

is

the

diameter of the

circle? (b)

27. (15)

what

the perimeter of the hexagon?

is

Find fractions equivalent of

6.

and

with denominators Subtract the smaller fraction you found from the to |

\

larger fraction. 28.

What type

i7)

(a)

ZRQS?

(b)

ZPQR?

(c)

ZPQS?

is

cups of water are poured from a container, how many ounces of water would be

29. If (16)

of angle

two

full

full

quart

left

in the

quart container? 30.

m

Find the perimeter of the hexagon

q

at right. 4

^

^ 6

in.

in.

Lesson 30

.

E S S

1

95

O N

30

Common Denominators Adding and Subtracting Fractions with Different

Denominators WARM-UP Facts Practice: Lines, Angles, Polygons (Test F)

Mental Math: a.

$1.99 + $1.99

d.

5

g.

Find

b.

$0.15 x 1000

if + 2§ 88, + 4, 8, x 5,

x 84

e.

\

>

of

c.

|

=

?

12

| of 20 that 5, double f.

-

number,

- 2,7 2,7 2,t2. Problem Solving:

Copy

this

problem and

fill

in the missing digits:

3_ _3)_6_6 1_

_0_

JEW CONCEPTS

Common When two denominators

have

have the same denominator, we say they denominators.

fractions

common 3

6

3

3

8

8

8

4

These two fractions do not have common denominators.

These two fractions have

common

denominators.

do not have common denominators, then one or both fractions can be renamed so both fractions do have common denominators. We remember that we can rename a fraction by multiplying it by a fraction equal to 1. Thus by multiplying by §, we can rename f so that it has a denominator of 8.

If

two

fractions

3

2

6

4

2

8

196

Saxon Math 8/7

Example

1

Solution

Rename

and

§

\ so that they

common

have

denominators.

The denominators are 3 and 4. A common denominator for these two fractions would be any common multiple of 3 and 4. The least common denominator would be the least common multiple of 3 and 4, which is 12. We want to rename each fraction so that the denominator 2 3

is 12.

1

12

12

4

We multiply \ by f and multiply \4 by

Thus

|

and

3 3*

2

4

_8^

1

3

3_

3

4

12

4

3

12

\ can be written with

— 12

common

denominators as

3_

and

12

with common denominators compared by simply comparing the numerators. written

Fractions

Example 2

common

Write these fractions with compare them.

6

Solution

The

least

of 6

and

9,

which

Then we

is

5

3

6

3

different

denominators

the

LCM

we may

write

fractions

15 18

7

2

is

9

2

14 18

we may

write y|, and in place of | compare the renamed fractions.

15 18

fractions with

9

18.

15 18

subtracting

be

denominators and then

common denominator for these

In place of |

Adding and

W

can

^

18

14 18

renamed compared

two fractions that do not have common denominators, we first rename one or both fractions so they do have common denominators. Then we can add or

To add

or subtract

subtract.

Lesson 30

Example

3

1

97

— + —

Add:

4

8

Solution

have common denominators are 4 and 8. The least

the ?

so

fractions

and

of 4

8

is

8.

they

We rename | so the We do not need to

by multiplying by add the fractions and simplify. 3

2

6

4

?

8

3

3

8

8

renamed |

9

added

8

- = 8

Example 4

bu:

t

a 8

.

6

Solution

Firs:

w

the

LCM

of 6

and 4

We

multiply | by f

we

::f

subtract th

10

2

77

=

3

:_

renamed |

ztk

renamed f

-7-

subtracted

12 12

2

1

3

6

Then

: ira;:::::?.

3

Solution

is 12.

common

so that both denominators are 12.

6

5

have

they

and multiply

2

Example

so

fractions

^uzrac:: 6- - 5-

W

write the fractions so that they have common >minators. The LCM of 3 and 6 is 6. We multiply f by § so first

the denominator is

6.

Then we

3

3 | =

and simplify.

renamed 8^

!-s| 3

subtract

1

2

subtracted and simplified

198

Saxon Math 8/7

Example 6 M Solution

12

3 Add: - + - + — 2

4

3

The denominators are 2, 3, and 4. The LCM of 2, 3, and 4 is 12. We rename each fraction so that the denominator is 12. Then we add and simplify. 1

2

'

2 '

3

3

+

4

'

6 _ 6 12 6

renamed

|

4 4

renamed

12

|

3

9

12

renamed

3

|

8

23 12

Example 7

added and

12

Use prime factorization

to

help you add these fractions:

24

32 Solution

We write the prime factorization of the

denominators

for

both

fractions. 5

5

32

2

•

2

•

2

7

7 •

2

•

24

2

2

•

2

2

•

•

3

common denominator of the two fractions is the least common multiple of the denominators. So the least common denominator is The

least

2-2-2-2-2-3=96 To rename multiply

5

32

the fractions with

by

3

3,

common

and we multiply

A 32

by

^

3

2

2 '

2

•

2

denominators,

we

2

15 96 28

3 '

_7_

24

7

.

96

2

43 96

LESSON PRACTICE Practice set*

Write the fractions so that they have

Then compare

3^7

common

the fractions. 5

r\

7

denominators.

Lesson 30

Add c.

1 + 5 + 1 6

d.

7^ - 2-

8

6

4- + 54

g.

99

or subtract:

4

e.

1

f.

2*

4

8

Use prime

2

6 factorization

to

help

9

you subtract these

fractions: 3_

_2_

25

45

MIXED PRACTICE Problem set

1. {28)

The

5 starters

on the basketball team were

tall.

Their

heights were 76 inches, 77 inches, 77 inches, 78 inches, and 82 inches. What was the average height of the 5 starters?

2. (28)

3. (29>

4. ll4)

Marie bought 6 pounds of apples for $0.87 per pound and paid for them with a $10 bill. How much did she get back in change? 317 rocks averaging 38 pounds each. He calculated that he had lifted over 120,000 pounds in all. Barney thought Fred's calculation was unreasonable. Do you agree or disagree with Barney? Why? Fred

lifted

One hundred

forty of the

the auditorium were boys. in the auditorium

5. (22>

two hundred

were

What

sixty students in

fraction of the students

girls?

following statement, change the percent to a reduced fraction. Then diagram the statement and answer In the

the questions.

The Daltons completed 30% of

their 2140-mile

trip the first day. (a)

(b)

How many miles did they travel the first day? How many miles of their trip do they still

have

to

travel?

the perimeter of a square is 5 S q Uare ? si(j e Q f | on g g

6. If (i6, i9)

.

feet,

how many

inches

200

Saxon Math 8/7

7.

Use prime factorization

to subtract these fractions:

(30)

1_

1

30

18 8.

(a)

Round 36,467

to the nearest

thousand.

(b)

Round 36,467

to the nearest

hundred.

(29)

9.

Mentally estimate the quotient

(29>

by

10.

(a)

Write

(b)

Use prime factorization

when

29,376

divided

is

49.

32%

as a

reduced

fraction.

(15, 24)

11. Write

m

these

fractions

to

so

reduce that

||.

they

common

have

denominators. Then compare the fractions.

6

W8

In the figure below, a 3-by-3-in. square

joined to a 4-by-4-in. square. Refer to the figure to answer problems 12 and 13. is

What

is

the area of the smaller square?

(b)

What

is

the area of the larger square?

(c)

What

is

the total area of the figure?

13. (a)

What

is

12. (a) (20)

L

>

(b)

the perimeter of the hexagon that by joining the two squares?

is

formed

The perimeter of the hexagon is how many inches less than the combined perimeter of the two squares? Why?

14. (a) Write the

prime factorization of 5184 using exponents.

(21)

(b)

15.

Use the answer

What

is

the

mean

to (a) to find

of

V5184.

5, 7, 9, 11, 12, 13, 24, 25, 26,

(28)

16. List the single-digit divisors of 5670. (6)

and 28?

Lesson 30

201

Solve: 17. (3.

6w

= 63

18. 90 c + 30° + a = 180°

20)

(3)

19. S45.00 =

36p

(3)

20.

—

(3)

32

= S3. 75

Simplify: 21. 21. (30)

23. T3o;

11

-1 + -

2

22.

3

(30;

5

1

6

2

2- - 1-

24. f26;

-3 -

-1

4

3

4 -1-18 2

5

3

25.

1-

2-

26.

3t1-7

f26j

4

3

(26)

8

2

3

1

For problems 27 and 28. record an estimated answer and an exact answer. 27. 33 (30) 29. lhlv 2>

+ 1-

28.

6

(23, 30)

Draw

5- - 18

4

with a compass, and label the center point O. Draw chord AB through point O. Draw chord CB not through point O. Draw segment CO. a

circle

30. Refer to the figure

drawn

(a)

Which chord

(b)

Which segments

(c)

Which

is

in

problem 29

to

answer

a diameter?

are radii?

central angle

is

an angle of AOBC?

(a)-(c).

202

Saxon Math 8/7

Focus on

Coordinate Plane By drawing two perpendicular number the tick marks,

we can

lines

and extending

create a grid over an entire plane

called the coordinate plane.

We can identify any point on the

coordinate plane with two numbers.

(3,2)

(-3, 2)

6 -5 -4 -3 -2 -1_0

1

2

3

4

5

6

-2 (-3, -2) -3

(3,-2)

-5

-6

The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The point at which the x-axis and the y-axis intersect is called the origin. The two numbers that indicate the location of a point are the coordinates of the

The coordinates

numbers in parentheses, such as (3, 2). The first number shows the horizontal direction and distance from the origin. The second number shows the vertical (t) direction and distance from the origin. The sign of the number indicates the point.

are written as a pair of

direction. Positive coordinates are to the right or up. Negative

coordinates are to the

left

or

down. The

origin

is at

point

(0, 0).

The two axes divide the plane into four regions called quadrants, which are numbered counterclockwise, beginning with the upper right, as first, second, third, and fourth. The

203

Investigation 3

signs of the coordinates of each quadrant are

Every point on a plane

shown below.

either in a quadrant or

is

on an

axis.

i

II

I

Second

First

quadrant

quadrant

(-.+)

(+, +)

6 -5 -4 -3 -2

-10

2

1

3

4

5

6

IV

Third

Fourth

quadrant

quadrant

(-.

-)

(+,-) i

Example

1

Find the coordinates

points A,

for

B,

and C on

this

coordinate plane.

5

B

H

6 -5 -4 -3 -2 -1_

1

2

3

4

5

6

—2 -3

If

Solution

We

find the point on the x-axis that is directly above, below, or on the designated point. That number is the first coordinate. Then we determine how many units above or first

below the

x-axis the point

is.

That number

coordinate.

Point

A

(4, 3)

Point

B

(-3, 4)

Point

C

(-5, 0)

is

the second

~

204

Saxon Math 8/7

Coordinate Plane

Activity:

Materials needed: •

Photocopies of Activity Master 8 (1 each per student; masters available in the Saxon Math 8/7 Assessments and Classroom Masters); graph paper may also be used

•

Straightedge

•

Protractor

We

suggest students work in pairs or in small groups. If using graph paper instead of Activity Master 8, begin by drawing an x-axis and y-axis by darkening two perpendicular lines on the graph paper. For this activity we will let the distance between adjacent lines on the graph paper represent a distance of one unit.

Example 2

Graph the following points on (a)

Solution

(3,4)

(b)

To graph each

a coordinate plane:

(2,-3)

we

point,

(-1,2)

(c)

begin

at the origin.

(d)

(0,-4)

To graph

(3,4),

we move to the right (positive) 3 units along the x-axis. From there we turn and move up (positive) 4 units and make a dot.

We

label the location (3,4).

We follow a similar procedure for

each point.

(3,4)

(-1,2) #2

-6-5-4-3-2-1

—1

—— —— I

I

-2

1

2

3

4

5

6

(2,-3)

3

\

-(0, -4) t -5

-6

Example 3

The and

vertices of a square are located at (-1, 2).

Draw

the square and find

(2, 2), (2,

its

-1), (-1, -1),

perimeter and area.

—

»

1

Investigation 3

Solution

We graph the vertices

205

and draw the square.

y

6 5

-4 3

(-"

'(2

i)

i

1

6 -5

-4 -3 -2

-(" -1,

3

1

)

-

-2

4

5

6

-1)

-3

-4 -5 -6

We

find that each side of the square

perimeter

Example 4

is

12 units, and

its

area

is

is 3

units long. So

its

9 square units.

Three vertices of a rectangle are located at (2, 1), (2, -1), and (-2, -1). Find the coordinates of the fourth vertex and the perimeter and area of the rectangle.

Solution

We graph the given coordinates. y

6

-5 -4 -3

-2 -1

6-5-4-3 ,

-( -2

1)

»

-? -2

— .

3 (2

>

4

5

6

1)

-3

-4 -5 -6

We see that the location of the fourth vertex is (-2, graph.

Then we draw

the rectangle and find that

1),

which we

it is

4 units

206

Saxon Math 8/7

long and 2 units wide. So

its

perimeter

is

12 units, and

its

area

is

8 square units.

(2,1)

(-2, 1)

-6 -5 -4 -3

(-2,-1 )._!;

1.

2.

3.

4

3

-1

(2,

5

6

-1)

Graph these three points: (2, 4), (0, 2), and (-3, -1). Then draw a line that passes through these points. Name a point in the second quadrant that is on the line. vertex of a square is the origin. Two other vertices are located at (-2, 0) and (0, -2). What are the coordinates of the fourth vertex?

One

Find the perimeter and area of a rectangle whose vertices are located at

(3,

-1), (-2, -1), (-2, -4),

and

(3,

-4).

4.

and (0, 0) are the vertices of a triangle. The triangle encloses whole squares and half squares on the grid. Determine the area of the triangle by counting the whole squares and the half squares. (Count two half squares as one square unit.)

5.

Draw a ray from the origin through the point (10, 10). Draw another ray from the origin through the point (10, 0). Then use a protractor to measure the angle.

6.

Points

Name (a)

7.

(4, 4), (4, 0),

the quadrant that contains each of these points:

(-15,-20)

(b)

(12,1)

Draw AABC with vertices Use a protractor triangle.

at

(c)

A

to find the

(20,-20)

(0, 0),

(d)

(-3,5)

B (8, -8), and C(-8, -8).

measure of each angle of the

Investigation 3

8.

207

Shae wrote these directions for a dot-to-dot drawing. To complete the drawing, draw segments from point to point in the order given.

9.

10.

1.

(0.4)

2.

(-3.-4)

3.

(5. 1)

4.

(-5. 1)

5.

(3.-4)

6.

(0.4)

Plan and create a straight-segment drawing on graph paper. Determine the coordinates of the vertices. Then write directions for completing the dot-to-dot drawing for other classmates to follow. Include the directions "lift pencil" between consecutive coordinates of points not to be connected.

Graph

a dot-to-dot design created

by

a classmate.

208

Saxon Math 8/7

LESSON 31

Reading and Writing Decimal Numbers

WARM-UP Facts Practice: +

-x

-r

Fractions (Test G)

Mental Math: ?

a.

$4.00 - 99C

d.

Reduce

g.

Start

with the number

t

-

5,

18 24

1,

35C

b.

7 x

e.

a/Too + 3

c. 2

12

60 I of degrees in a right angle, f.

of

2,

+

5,

find the square root.

Problem Solving: Find the next three numbers in

this sequence:

100, 121, 144,

...

NEW CONCEPT We

have used fractions and percents to name parts of a whole. We remember that a fraction has a numerator and a denominator. The denominator indicates the number of equal parts in the whole. The numerator indicates the number of parts that are selected.

Number of parts selected = Number of equal parts in the whole

Parts of a

whole can

also be

In a decimal fraction

— 10 3

ITTT

named by using decimal fractions.

we can

see the numerator, but

we

cannot

The denominator of a decimal fraction is indicated by place value. Below is the decimal fraction three see the denominator. tenths. is

We know the denominator is

shown

to the right of the

10 because only one place

decimal point. 0.3

J

Lesson

The decimal

fraction 0.3

31

209

and the commo

equivalent. Both are read '"three tenths."

0.3

=

10

A

decimal fraction written with two c point (two decimal places) is un denominator of 100. if we show heir: 0.03 =

100 0.21

= 100

A number that contains number

a

:

.

decimal

or just a decimal, i

(W

7

12

fraction:

=

?

24

> f (C)

?

6

_ 4

" 24

prime factorization of 2025 using exponents.

(21)

(b)

-3), (-3, -3),

Find a/2025.

214

Saxon Math 8/7

15. ll8}

Draw two

parallel lines.

Then draw two more

paralle

Labe

lines that are perpendicular to the first pair of lines.

D consecutively ii

the points of intersection A, B, C, and a counterclockwise direction. the figure to answer (a)

What kind

(b)

Triangles

(a)

and

ABC

Refer

t(

(b):

of quadrilateral

and CDA

AABC corresponds

in

Draw segment AC. figure

is

ABCD?

are congruent.

Which

angl


2|,

double both numbers

and then find the quotient.

12. (a) Write (5,

by

2500 in expanded notation.

21)

(b)

Write the prime factorization of 2500 using exponents.

(c)

Find V2500.

13. If 35 liters of petrol cost $21.00,

what

is

the price per liter?

(13)

14. (17>

Use a protractor and a 45° angle.

to

draw

a triangle that has a 90° angle

In the figure below, a 6-by-6-cm square

square. Refer to the figure to

is

joined to an 8-by-8-cm

answer problems 15 and

What

is

the area of the smaller square?

(b)

What

is

the area of the larger square?

(c)

What

is

the total area of the figure?

15. (a)

16.

(20)

16.

What

the perimeter of the hexagon that e the squares joining squares?9 is

is

formed by

j

Solve: 17. 10 (3)

6 =

Aw

18. 180° (3)

-

s

= 65°

Lesson 32

221

oimpiiiy.

(30;

21 (30)

23. fZ3, 30)

+ 1 + 7 8 4 -5

*L

16

20

1

20.

2

(30)

22.

24.

2

a

25f25j

3

6

4

8

li. 10

9

1 1 6^ - 2^

D

5

+ fl * ] ! 3 \3 2j

5

4^ +

(30)

8

26.

25 36

G?4

a2 9

8

10

15

For problems 27 and 28, record an estimated answer and an exact answer:

(26;

29. (blv3)

5

— 10 9

2

27. 5-

-r

28. f30j

The coordinates of three (-5, -2), and (2, -2).

3 7 7- + 1-

4

8

vertices of a rectangle are (-5,

(a)

What

are the coordinates of the fourth vertex?

(b)

What

is

the area of the rectangle?

30. Refer to the figure

below

to

answer

(a)-(c).

(Inv. 2)

c

(a)

Which chord

(b)

Name

(c)

Name an

is

not a diameter?

a central angle that

inscribed angle.

is

a right angle.

3),

222

Saxon Math 8/7

LESSON

33

Comparing Decimals Rounding Decimals

WARM-UP Facts Practice: Lines, Angles, Polygons (Test F)

Mental Math: a.

$2.84 - 99(2

b.

d.

Reduced. 30'

e.

g.

6 x 550 2

c.

- V25

3

_

?

24

'"6 of 30 Think of an equivalent division problem for 600 -r w 5

f.

*

50.

Then

find the quotient.

Problem Solving:

When Bill, Phil, Jill, and Gil entered the room, they all shook hands with each other. How many handshakes were there in all? Diagram the situation. (Four students may act out the story.)

NEW CONCEPTS Comparing decimals

Example

1

When comparing

decimal numbers, it is necessary to conside] place value. The value of a place is determined by its positior with respect to the decimal point. Aligning decimal pointf can help to compare decimal numbers digit by digit.

Arrange these decimal numbers in order from 0.13

Solution

We

0.0475

0.128

will align the decimal points

column by column.

First

we

look

least to greatest

at

and consider the

digits

the tenths place.

0.13

0.128 0.0475

Two

numbers have a 1 in the tenths place number has a 0. So we can determine thai

of the decimal

;

while the third 0.0475 is the least of the three numbers. Now we look at the hundredths place to compare the remaining two numbers.

0.13

0.128

Lesson 33

223

Since 0.128 has a 2 in the hundredths place, it is less than 0.13, which has a 3 in the hundredths place. So from least to greatest the order is 0.0475, 0.128, 0.13

Note that terminal zeros on a decimal number add no value the decimal number. 1.3

to

= 1.30 = 1.300 = 1.3000

When we compare two

decimal numbers, it may be helpful to insert terminal zeros so that both numbers will have the same number of digits to the right of the decimal point. We will practice this technique in the next few examples.

Example 2 Solution

Compare: 0.12O0.012 So that each number has the same number of decimal places, we insert a terminal zero in the number on the left and get 0.120

O 0-012

One hundred twenty thousandths thousandths, so

we

is

greater than twelve

write our answer this way: 0.12 > 0.012

Example 3 Solution

Compare:

We

0.4

O 0.400

can delete two terminal zeros from the number on the

right

and

get 0.4

We

= 0.4

could have added terminal zeros to the number on the

left to get

0.400 = 0.400

We write

our answer this way: 0.4 = 0.400

Example 4 Solution

Compare: 1.232

O 1.23185

We insert two terminal zeros in the number on the left and get 1.23200 Since 1.23200

is

O 1-23185

greater than 1.23185,

we

1.232 > 1.23185

write

Saxon Math 8/7

Rounding decimals

Example

5

Solution

To round decimal numbers, we can use the same procedure that we use to round whole numbers.

Round 3.14159

The hundredths place point.

hundredth.

to the nearest is

We underline the

two places

to the right of the

digit in that place

and

decimal

circle the digit

to its right.

3.14(1)59

Since the circled digit

than

we

leave the underlined replace the circled digit and all

is less

5,

unchanged. Then we digits to the right of it with zeros. digit

3.14000

decimal point do not serve as placeholders as they do in whole numbers. After rounding decimal numbers, we should remove terminal zeros to the right of the decimal point.

Terminal zeros

to the right of the

3.14000

i

i

GDGDGDGD G3G3GDGD

CD CO (X) CD CD CD CD CD CD CD CD CD CD CD GD CD

Note

that

a

calculator

—

3.14

simplifies

decimal

numbers by

omitting from the display extraneous (unnecessary) zeros. For example, enter the following sequence of keystrokes:

Notice that all entered digits are displayed. Now press the HI key, and observe that the unnecessary zeros disappear from the display.

Example 6 Solution

Round 4396.4315

We

to the nearest

hundred.

are rounding to the nearest hundred, not to the nearest

hundredth.

43©6.4315 Since the circled digit is 5 or more, we increase the underlined digit by 1. All the following digits become zeros.

4400.0000

end of the whole-number part are needed as placeholders. Terminal zeros to the right of the decimal point are not needed as placeholders. We remove these zeros. Zeros

at the

4400.0000

—

4400

Lesson 33

Example 7 Solution

Round

38.62 to the nearest whole number.

To round

a

number to

the nearest

the ones place. 38.(6;2

Example 8

Solution

225

—

whole number, we round

39.ee

—

to

39

Estimate the product of 12.21 and 4.9 by rounding each number to the nearest whole number before multiplying.

We

round 12.21 to 12 and 4.9 to 5. Then we multiply 12 and and find that the estimated product is 60. (The actual

5

product

is

59.829.)

ESSON PRACTICE Practice set*

Compare: a.

b.

c.

O 5.06 O 5.60 O 1-099 10.30

10-3

1.1

d.

Round 3.14159

e.

Round 365.2418

f.

g.

h.

Round 57.432

to the nearest ten-thousandth.

to the nearest

to the nearest

hundred.

whole number.

Simplify 10.2000 by removing extraneous zeros. Estimate the sum of 8.65, 21.7, and 11.038 by rounding each decimal number to the nearest whole number before adding.

MIXED PRACTICE Problem set

1. l28)

new school record when she cleared How can we find how many inches high

The high jumper 5 feet 8 inches.

set a

5 feet 8 inches is?

2. (28)

During the

first

week

of

November

the

daily

high

temperatures in degrees Fahrenheit were 42°F, 43°F, 38°F, 47°F, 51°F, 52°F, and 49°F. What was the average daily high temperature during the first week of

November?

226

Saxon Math 8/7

3. 111

4.

In 10 years the population increased

from 87,196

120,310. By how many increase in 10 years?

the

people

Find the next two numbers in

did

to

population

this sequence:

(2)

120, 105, 90, 75,

5. ig)

6.

A

and

regular hexagon

...

a regular

octagon share a common side. If the perimeter of the hexagon is 24 cm, what is the perimeter of the octagon?

Diagram

statement.

this

Then answer

the questions

'

22]

that follow.

One

third of the 60 questions

on the

test

were

true-false. (a)

How many

of the questions

on the

test

were

true-

false? (b)

How many

of the questions

on the

test

were not

true-

false? (c)

7. 3)

8.

What percent

of the questions

were

true-false?

Find the area of a square whose vertices have the coordinates (3, 6), (3, 1), (-2, 1), and (-2, 6). (a)

Round 15.73591

(b)

Estimate

to the nearest

hundredth.

(33)

product of 15.73591 and 3.14 by rounding each decimal number to the nearest whole

number 9.

Use words

the

before multiplying.

to write

each of these decimal numbers:

(31)

10.

(a)

150.035

(b)

0.0015

Use

digits to write

each of these decimal numbers:

(31)

(a)

one hundred twenty-five thousandths

(b)

one hundred and twenty-five thousandths

Lesson 33

each circle with the proper comparison symbol:

11. Replace (331

12.

O

0.128

(a)

227

O-l 4

Q 0.0015

0.03

(b)

Find the length of this segment

(32)

cm

1

I

(a)

in centimeters.

(b)

in millimeters.

13.

Draw

(17)

draw

15. (27>

3

4

I

I

I

5

L

the straight angle AOC. Then use a protractor to ray OD so that angle COD measures 60°.

we

multiply one integer by another integer that is a whole number but not a counting number, what is the product?

14. If 121

2

Use prime

factorization

to

the

find

least

multiple of 63 and 49.

Solve: 16.

8m

= 4

•

18

17. 135°

+ a = 180°

(3)

(3)

Simplify: 18. (30)

20. 00)

22. (26)

24. (23, 30;

- + - + 4

8

2

4- — — 2

8

2^- - 5| 10

26. $40.00

4

21.

—

(26)

8

6

•

25. (25j

16

2—

4-

+ j 4

•

3— 3

5

(26)

6

-r

- - |

(30)

23. 5

5

6^ - 2§ 2

19.

4D 1

2

v2

3

common

228

Saxon Math 8/7

27.

(a)

Solve: 54 = 54 +

(b)

What property

y

(2)

Consider

29.

When

1291

by the equation in

(a)?

following division problem. WithoU dividing, decide whether the quotient will be greater thar 1 or less than 1. How did you decide?

28.

m

illustrated

is

the

Kelly saw the following addition problem, she knew that the sum would be greater than 13 and less thar 15. How did she know?

si**! 3 8 30. (17)

Use a protractor and a 60° angle.

to

draw

a triangle that has a 30° angle

i

Lesson 34

2 29

.ESSON Decimal Numbers on the

Number Line

(VARM-UP Facts Practice: +

—

x

-f

Fractions (Test G)

Mental Math: a.

$6.48 -

d.

Reduce g.

g.

98(2

b.

5 x

48c

e.

a/36

•

^49

c. f.

f

=

^

|of36

Square the number of sides on a pentagon, double that number, - 1, V~, x 4, - 1, 3,

Problem Solving: Jamaal glued 27 small blocks together to make this cube. Then he painted the six faces of the cube. Later the cube broke apart into 27 blocks. How many of the small blocks had 3 painted faces? ... 2 painted faces? ... 1 painted face? ... no painted faces?

slEW

CONCEPT If

the distance between consecutive

whole numbers on

a

number line is divided by tick marks into 10 equal units, then numbers corresponding to these marks can be named using decimal numbers with one decimal place. An example of this kind of number line is a centimeter ruler. each centimeter segment on a centimeter scale is divided into 10 equal segments, then each segment is 1 millimeter long. Each segment is also one tenth of a centimeter long.

If

Example

1

Solution

Find the length of (a)

in millimeters.

(b)

in centimeters.

(a)

Each centimeter the scale

(b)

is 1

segment

this

mm.

mm.

Thus, each small segment on The length of the segment is 23 mm.

is

10

Each centimeter on the scale has been divided into 10 equal parts. The length of the segment is 2 centimeters plus three tenths of a centimeter. In the metric system we use decimals rather than common fractions to indicate parts of a unit. So the length of the segment is 2.3 cm.

— 230

j

Saxon Math 8/7

the distance between consecutive whole numbers on number line is divided into 100 equal units, then number If

corresponding to the marks on the number line can be namei using two decimal places. For instance, a meter is 100 cm. S each centimeter segment on a meterstick is 0.01 (or ^) of th length of the meterstick. This means that an object 25 cm Ion is

Example 2

m long.

also 0.25

Find the perimeter of

this rectangle

7cm

in meters. 12

Solution

The perimeter is

^

of a meter. So 38

as 0.38

Example 3

of the rectangle

^

is

38 cm. Each centimete

of a meter,

ABC

m.

\

\

mi

i

i

l

4.0

We

ii +i ii

i

l

ii

4.1

Mlm

i

li

i

Hlim mi

4.2

i

ii

\

4.5

4.6

4.7

4.8

number

4.9

5.0

from 4 to I The distance from 4 to 5 has been divided into 100 equa segments. Tenths have been identified. The point 4.1 is on tenth of the distance from 4 to 5. However, it is also tei hundredths of the distance from 4 to 5, so 4.1 equals 4.10. are considering a portion of the

Arrow A

indicates 4.05.

Arrow B

indicates 4.38.

line

4.73.

Find the following sum (a)

in millimeters.

(b)

in centimeters. 4.2

Solution

—

wiH 4.4

4.3

Arrow C indicates Example 4

which we writ

Find the number on the number line indicated by each arrow

— Solution

cm

is

cm

cm

+ 24

mm

(a)

We express 4.2 cm as 42 mm

(b)

mm + 24 mm = 66 mm We express 24 mm as 2.4 cm and add.

and add.

42

4.2

cm

+ 2.4

cm

= 6.6

cm

231

Lesson 34

ESSON PRACTICE Practice set

Refer to the figure below to answer problems a-c. cm

2

1

I

I

a.

Find the length of the segment in centimeters.

b.

Find the length of the segment

c.

What

to the nearest millimeter.

the greatest possible error of the

is

problem b? Express your answer

as

measurement a

fraction

in

of a

millimeter. d.

Seventy-five centimeters

e.

Alfredo is 1.57 meters Alfredo?

f.

What point on and

g.

a

is

tall.

number

how many

meters?

How manv line

is

centimeters

tall is

halfway between 2.6

2.7?

What decimal number names number line?

the point

marked

A

on

this

A 10.0

h.

10.1

Estimate the length of this segment in centimeters. Then use a centimeter ruler to measure its length.

1.

3.5

j.

4

cm

cm -

+ 12 12

mm

mm

=

=

cm

mm

__

MIXED PRACTICE Problem set

1. l28)

2. (11>

In 3 boxes of cereal, Jeff

counted 188

raisins,

212 raisins,

and 203 raisins. What was the average number in each box of cereal?

of raisins

The pollen count had increased from 497 parts per million to 1032 parts per million. By how much had the pollen count increased?

232

Saxon Math 8/7

3.

Sylvia spent $3.95 for lunch but still had $12.55. much money did she have before she bought lunch?

How

1903 the Wright brothers made the first powerec airplane flight. Just 66 years later astronauts first landec on the Moon. In what year did astronauts first land oi

4. In 1121

the

5. ll9)

Moon?

The perimeter

of the square equals

the perimeter of the regular hexagon.

each side of the hexagon is 6 inches long, how long is each side of If

the square?

6. t22)

In the following statement, write the percent as a reducec

Then diagram

fraction.

the statement and answer

th(

questions.

week

Each

Jessica

saves

40%

of her $4.00

allowance. (a)

How much

allowance

money does

she save eacl

week? (b)

How much

allowance money does she not save eacl

week?

7.

Describe

how to

estimate the product of 396 and 71.

(29)

8.

Round 7.49362

to the nearest thousandth.

(33)

9.

Use words

to write

each of these decimal numbers:

(31)

10.

(a)

200.02

(b)

0.001625

Use

digits to write

each of these decimal numbers:

(31)

(a)

one hundred seventy-five millionths

(b)

three thousand, thirty

and three hundredths

—

1

1

2 33

Lesson 34

11.

Replace each circle with the proper comparison symbol:

(33)

(a)

12.

6.174

O 6.17401

Q 1.4276

14.276

(b)

Find the length of this segment

cm

2

1

(34)

13. (34}

(a)

in centimeters.

(b)

in millimeters.

iniliiiiliiiil

What decimal number names number line?

the point

1

1

1

1

1

1

1

marked

x

——

+

|

i

1

1

1

1

1

1

1

14. -

1

1

of three vertices of a square are

are the coordinates of the fourth vertex?

(b)

What

is

What decimal number

is

What number

20 = 12y

is

halfway between 0.7 and 0.8?

17. 180°

= 74° + c

(3)

(3)

Simplify:

00)

20.

5 + - + \

19.

2

00)

36

5^ - \\

21.

3

(26)

^ lU

l|

23.

3

(26)

6

3

(23,30)

6

22.

5-

(26)

4

24. (30)

"T*

6^ + 4±

25.

4

(9,24)

8

1

(0, 0),

halfway between 7 and 8?

Solve:

18.

1

the area of the square?

(34)

•

V

—

What

16. 15

1

X on this

(a)

(b)

1

8.3

The coordinates (0, 3), and (3, 3).

15. (a)

1

1

8.2

(Inv 3)

1

3^

24

•

2§ O

-r

•

3f 4

4

O

+ I 8

V6

4,

234

Saxon Math 8/7 26. Express the following difference (a) in centimeters (34)

(b)

in millimeters. 3.6

27.

and

Which

is

cm -

24

mm

3

2 equivalent to 2

•

2 ?

(20)

A. 2 5 28.

B. 2

6

D. 24

C. 12

Arrange these numbers in order from

least to greatest:

(33)

0.365, 0.3575, 0.36

29. Evaluate this expression for

x =

5

and y -

10:

(l, 4)

x 30. nv 21

Use

a

draw three concentric l| inches, and 2 inches.

compass

radii 1 inch,

to

circles.

Make

the

.

2 35

Lesson 35

LESSON

35

Adding, Subtracting, Multiplying, and Dividing Decimal Numbers

/VARM-UP

Facts Practice: Measurement Facts (Test H)

Mental Math: a.

d. g.

$7.50 - $1.99

Reduce Start

b. e.

24

x 64tf

5

15

4

2

-

c.

A/4

f.

with the number of inches in two

What do we

call this

many

10

~

30

A of 24

feet,

+

x 4, "f~

1,

years?

Problem Solving:

Copy

this

problem and

fill

R5

in the missing digits:

8

16 24

MEW CONCEPTS Adding and subtracting

decimal

numbers Example

1

Solution

Adding and subtracting decimal numbers is similar to adding and subtracting money. We align the decimal points to ensure that we are adding or subtracting digits that have the same place value.

Add:

3.6 + 0.36

+ 36

We align the decimal points vertically. A number written without a decimal point

is

a

whole

decimal point

is

number,

so

to the right of 36.

the

3.6

0-36

+ 36 39.96 -

236

Saxon Math 8/7

Example 2 Solution

Add:

We and

0.1

+ 0.2 + 0.3 + 0.4

align the decimal points vertically add. The sum is 1.0, not 0.10.

Since 1.0 equals

answer

Example

3

Solution

1,

we can

0.1 0.2

simplify the

0.3

+ 0.4

to 1.

Subtract: 12.3

- 4.567

We

first

number above

i

the

o

second number, aligning the decimal

1

write the

points.

We

write zeros in the

-

empty

places and subtract.

Example 4 Solution

Subtract: 5

2 9

a

2.3^0 4.567 7.733

- 4.32

We

write the whole number 5 with a decimal point and write zeros in the

two empty decimal

places.

4

9

4.3

Then we

i

2

0.68

subtract.

Multiplying decimal

1

we

multiply the fractions three tenths and seven tenths, the product is twenty-one hundredths, If

numbers 3_ x

io

7_

21

10

100

we

multiply the decimal numbers three tenths and seven tenths, the product is twenty-one hundredths. Likewise,

if

0.3 x 0.7 = 0.21

Here

we

use an area model to illustrate this multiplication:

1

13. 01,35)

(a)

with a bar over the repetend.

(b)

rounded

to three

decimal places.

Four and five hundredths is how much greater than one h unc rec sixty-seven thousandths? i

i

302

Saxon Math 8/7

14.

Draw AB

perpendicularjo AB. Complete

How 15.

m

long

AC

inch__hmg. Then draw

1

AABC

f

inch

l

ong

by drawing BC.

BC?

is

normal deck of cards is composed of four suits (red heart, red diamond, black spade, and black club) of 13 cards each (2 through 10, jack, queen, king, and ace) for a total of 52 cards. If one card is drawn from a normal deck of cards, what is the probability that the card will be a red

A

card?

16. °-

(a)

21)

17. 1321

Make

showing the prime factorization with the factors 30 and 30.)

a factor tree

900. (Start

of

(b)

Write the prime factorization of 900 using exponents.

(c)

Write the prime factorization of a/900

The eyedropper held

.

2 milliliters of liquid.

eyedroppers of liquid would

take to

it

How many

fill

a

1 -liter

container?

18. (a)

8%

Write

as a

decimal number.

(43)

(b)

Find to

19.

(a)

{19 ' 37)

8%

(a).

What

by multiplying $8.90 by the answer Round the answer to the nearest cent. is

of $8.90

the perimeter of this

triangle? (b)

What

0.6

the

is

area

of

m

this

0.8

triangle?

20.

Compare and explain

the reason for your answer:

(27)

32 2

21. Evaluate a[b

+

c]

if

320 q ^ 20

a =

2,

b =

3,

and c =

4.

(41)

Solve:

22 09)

— 18

_iL

~ 4.5

23. 1.9 (35)

=

w

+ 0.42

m

Lesson 43

303

Simplify: 24. 6.5

-

"J

^5;

1

26. 5- + caoj

28.

25.

3—

(23, 30)

10

4

q 6—

+

10

2

4 -

27.

5

1

15

114

7-

•

3-

2

Find the next coordinate pair in

11

-

•

-

3

-r

5

5

this sequence:

(Inv. 3)

(1,2), (2,4), (3,6), (4, 8),

29.

m

Find the measures of Za, Zb, and

Zc in

30. Refer nv 2)

...

the figure at right.

answer (a)

(b)

the

to

figure

right

to

(a)-(c):

What

is

angle

AOB?

the measure of central

What appears

to

be the measure

of inscribed angle (c)

at

Chord

AC is

ACB?

congruent to chord BC. What appears to

be the measure of inscribed angle

ABC?

304

Saxon Math 8/7

LESSON Division Answers WARM-UP Facts Practice: Measurement Facts (Test H)

Mental Math: a.

5 x

0.5

b.

64$

-r

10

d. Estimate:

| of

f.

g.

Start

-

1,

596

11

200

with the number of meters in a kilometer, V~, x 5, + 1, aT.

-f

10,

,

x 5,

Problem Solving:

The prime number numbers?

7 is the

average of which two different prime

NEW CONCEPT We can write answers to division problems with remainders in different ways. We can write them with a remainder or as a mixed number.

6R3 4j27 24

4j27 24

3

3

We

can also write the answer as a decimal number. We fix place values with a decimal point, affix zeros to the right of the decimal point, and continue dividing. 6.75

4)27.00 24 3

2 8

20 20

Lesson 44

Example

1

Solution

305

Divide 54 by 4 and write the answer (a)

with a remainder.

(b)

as a

(c)

as a decimal.

(a)

(b)

mixed number.

We divide 13 R 2.

result

We

2

4j54

1^ 2

to 13|.

place values by placing the decimal point to the right of 54. Then we can write zeros in the following places and continue dividing until the remainder is

13.5

fix

zero.

R

13

is

The remainder is the numerator of a fraction, and the divisor is the denominator. Thus this answer can be written as 13|, which reduces

(c)

and find the

The

result

is

4)54.0 4

14 12

~2

13.5.

q

2

number will be a repeating decimal number or will have more decimal places than the problem requires. In this book we show the complete division of the number unless the Sometimes

problem Example

2

Solution

a division

states that the

answer written

answer

is

to

as a decimal

be rounded.

Divide 37.4 by 9 and round the quotient to the nearest thousandth.

We

continue dividing until the answer has four decimal places. Then we round to the nearest thousandth. 4.155 5

...

—

4.156

4.1555...

9)37.4000...

36 1

4 9

50 45 50 45 50 45 5

306

Saxon Math 8/7

Problems involving division often require us to interpret the results of the division and express our answer in other ways. Consider the following example.

Example

3

Solution

Vans will be used to transport 27 students on a Each van can carry 6 students. (a)

How many vans

(b)

How many vans will be

(c)

If all

can be

filled?

needed?

but one van will be full, then how will be in the van that will not be full?

The quotient when 27

is

field trip.

many

students

divided by 6 can be expressed in

three forms.

4R3

4| 6j27

6)27

The questions require us

to

4.5

6)27.0

interpret the

results

of the

division. (a)

The whole number 4 can be

in the quotient

means

that 4 vans

filled to capacity.

(b)

Four vans will hold 24 students. Since 27 students are going on the field trip, another van is needed. So 5 vans will be needed.

(c)

The

fifth

van will carry the remaining 3 students.

LESSON PRACTICE Practice set

Divide 55 by 4 and write the answer a.

with a remainder.

b. as a

mixed number. decimal number.

c.

as a

d.

Divide 5.5 by 3 and round the answer to three decimal places.

e.

Ninety-three students are assigned to four classrooms as equally as possible. How many students are in each of the four classrooms?

3 07

Lesson 44

MIXED PRACTICE Problem set

1. 1361

2. (28)

3. 1141

4. (28)

5. 01,35)

6. (22 36) '

The

rectangle

What was

was 24 inches long and 18 inches wide.

the ratio of

its

length to

its

width?

Lakeisha's test scores were 90, 95, 90, 85, 80, 85, 90, 80, 95, and 100. What was her mean (average) test score?

The

report stated that

two out of every

five

were unable to find a job. What fraction people were able to find a job?

young people of the young

Rachel bought a sheet of fifty 34-cent stamps from the post office. She paid for the stamps with a $20 bill. How much money should she get back? Ninety-seven thousandths is how much less than two and nme ty_ e ight hundredths? Write the answer in words.

Diagram

this

statement.

Then answer

the questions

that follow.

Five sixths of the 30 students passed the

7. {19)

(a)

How many

(b)

What was

students did not pass the test?

the ratio of students who passed the test to students who did not pass the test?

Copy

on your paper. Find the length of each unmarked side, and find the perimeter of the polygon. Dimensions are in meters. this

figure

All angles are right angles.

8.

test.

18

15

16

(a)

Write 0.75 as a reduced fraction.

(b)

Write | as a decimal number.

(c)

Write

(43)

9. (36)

125%

as a decimal

number.

normal deck of cards, what the probability that the card will be a heart? If

a card is

drawn from

a

is

308

Saxon Math 8/7

10.

which

2(3 + 4) equals

The expression

of the following?

(41)

A. (2 C.

11.

•

+ 4

3)

B. (2

+

(2

4)

•

D. 23 + 24

+ 7

2

3)

•

Find the next three numbers in

this sequence:

(2)

10,

1, 3, 6,

...

by 11 and write the answer

12. Divide 5.4 (42. 44)

13.

(a)

with a bar over the repetend.

(b)

rounded

What composite number

1211

first

equal to the product of the

is

four prime numbers?

Arrange these numbers in order from least to

14. (a) (4.

thousandth.

to the nearest

greatest:

33)

1.2,-12, 0.12,

Which numbers

(b)

15.

Each math book

is

in

(a)

0, \

are integers?

\\Z inches thick.

(26)

A

(a)

stack of 12

inches

math books would stand how many

tall?

How many math

(b)

books would make a stack

1

yard

tall?

16.


19.

^

(39J

8

a

12

w

-

20. 4.7

= 1.2

21.

lOx = 10 2

20)

(3,

(35)

100

Estimate each answer to the nearest whole number. Then

perform the calculation. 22.

11 + 2^4 24 18

11 1™

(30)

23.

3-1-1^

5^ -

(30)

D

25.

6^

3j

Simplify: 24. f26j

2

1

3

8

§ x 4 x i£

26. 3.45 + 6 + (5.2 28. Describe

how

-

27. 2.4

0.57)

to estimate the

-r

4

3

f26j

-f

0.016

product of 6| and 5^.

(29)

In the figure below, figure for

29. (18}

m

problems 29 and

congruent to ACDA. Refer to the

30.

the angle or side in AABC that corresponds to the following angle and side from ACDA:

Name (a)

30.

AABC is

ZACD

The measure is

60°.

(a)

(b)

of

ZACB

is

45°,

DC

and the measure of

Find the measure of

ZB.

(b)

ZCAB.

(c)

ZCAD.

ZADC

Lesson 47

323

LESSON

47

Powers of 10

A/ARM-UP

Facts Practice: Proportions (Test

I)

Mental Math: x $8.20

a.

5

C.

10

c

~

2

b.

9 15

d. Estimate:

3

f

| of

$4.95 x 19

60

e.

2

g.

Find the sum, difference, product, and quotient of \ and

•

2

0.015 x 10 3

.

|.

Problem Solving: chickens can lay a total of 2 eggs in eggs can 4 chickens lay in 4 days?

If 2

2 days,

then

how many

JEW CONCEPTS Place value as

powers of 10

The

positive

powers of 10 are easy

to write.

The exponent

matches the number of zeros in the product. 1(T = 10

10 3 = 10 10 4 = 10

•

10 = 100

•

10

•

10 = 1000

•

10

•

10

•

(two zeros)

10 = 10,000

(three zeros)

(four zeros)

Notice that when we multiply powers of 10, the exponent of the product equals the sum of the exponents of the factors. 6 3 10 3 x 10 = 10

1000 Also,

x

1000 = 1,000,000

when we

quotient

divide powers of 10, the exponent of the equals the difference of the exponents of the

dividend and divisor. 10 1,000,000

fi

-r

-r

10 3 = 10^

1000 = 1000

324

Saxon Math 8/7

We can use powers

of 10 to show place value, as chart below. Notice that 10° equals 1.

see in the

•*—/

Thousands

Millions

Billions

Trillions

we

Units (Ones)

c o

a eds

eds

'eds

hundi

hundi

hundi

14

10 13

10

12

11

10 9

10 10

hund

tens

tens

10

10 8

10

7

10 6

10 5

Powers of 10 are sometimes used expanded notation.

Example

1

Solution

Solution

tens

4

10 3

10 2

write

to

The number 5206 means 5000 + 200 + 6. each number as a digit times its place value.

(5

Example 2

10

Decin

ones

tens

10 1

ones

10°

numbers

in

Write 5206 in expanded notation using powers of 10.

5000

Multiplying by powers of 10

15

hundi ones

ones

ones

tens

10

'eds

reds

200

+

+

We

will write

6

x 10 3 ) + (2 x 10 2 ) + (6 x 10°)

When we

multiply a decimal number by a power of 10, the answer has the same digits in the same order. Only their place values are changed.

Multiply: 46.235 x 10 2

This time

we

will write 10 2 as 100

and multiply.

46.235 x

100 4623.500 = 4623.5

We

same digits occur in the same order. Only the place values have changed as the decimal point has been shifted two places to the right. To multiply a decimal number by a positive power of 10, we shift the decimal point to the right the number of places indicated by the exponent. Example 3

see that the

Multiply: 3.14 x 10 4

325

Lesson 47

Solution

The power

of 10

shows us the number of places

decimal point to the

right.

We move

to

move

the

the decimal point four

places to the right. 4 3.14 x 10 = 31,400

Sometimes powers of 10

are written with

words instead of

For example, we might read that 1.5 million spectators lined the parade route. The expression 1.5 million

with

digits.

means Example 4 Solution

which

1.5 x 1,000,000,

is

1,500,000.

Write 2| billion in standard form. First

we

write 2| as the decimal

multiply by one billion (10

9 ),

which

number JZ. 5. Then we shifts the

decimal point

9 places to the right. 2.5 billion

Dividing by powers of 10

= 2.5 x 10 9 = 2,500,000,000

When

dividing by positive powers of 10, the quotient has the same digits as the dividend, only with smaller place values.

4.75

-l.

io

3

To divide a number by a

Example

5

Solution

decimal point the exponent.

to the left

Divide: 3.5

10 4

-f

The decimal point decimal point in

—^

0.00475 1000)4.75000

power of 10, we shift the the number of places indicated by positive

of the quotient

is

4 places to the

left

of the

3.5.

3.5

-f

10 4 = 0.00035

ESSON PRACTICE Practice set

Write each number in expanded notation using powers of 10: a.

456

b.

1760

c.

186,000

— 326

Saxon Math 8/7

Simplify: d.

f.

3 24.25 x 10

12.5

-r

10

3

e.

6 25 x 10

g.

4.8

-r

10 4

i.

10 8

-f

10 2 = 10 a

Find each missing exponent: h.

10

3 •

10

- 10 a

4

Write each of the following numbers in standard form: j.

k. 15 billion

2^ million

1.6 trillion

1.

MIXED PRACTICE Problem set

Refer to the graph to answer problems 1-3.

How 16

Students

Come to

School

| 1

14

w c 12

,

I

CD

3

10

| 1

8

|

6

-,

|

E

1

I

2 o

—

I

1

I

—

Walk

1.

Answer

1

4

L_

Ride a Bike

1

—

I

1

—

1

I

Ride in a Car

Ride in a Bus

true or false:

(38)

(a)

Twice

as

many

school in a (b)

2. 381

3. (38)

What

who

is

of the students ride to school in either a

car.

the ratio of those

who walk

to

school to those

ride in a bus?

What

to

car.

The majority bus or

students walk to school as ride

fraction of the students ride in a bus?

327

Lesson 47

4.

What

the

is

mean

(average) of these

numbers?

(28, 35)

1.2, 1.4, 1.5, 1.7, 2

5.

(a)

The newspaper reported that 134.8 million viewers watched the Super Bowl. Write the number of viewers in standard form.

(b)

6. (22)

Write 5280 in expanded notation using powers of 10.

Diagram

this

Then answer

statement.

the questions

that follow.

Only one eighth of the 40 students correctly answered question 5. (a)

How many

(b)

How many

students correctly answered question 5? students

did

not

answer

correctly

question 5?

7. (44)

8. 181

A

gallon of

glasses.

punch (128 ounces)

is

poured into 12-ounce

\

(a)

How many glasses

(b)

How many glasses are needed to hold all of the punch?

A

can be

filled to the top?

an ancient unit of measure equal distance from the elbow to the fingertips. (a)

cubit

Estimate the

your (b)

is

number

to

the

of inches from your elbow to

fingertips.

Measure the distance from your elbow

your

to

fingertips to the nearest inch.

9.

(a)

Write 0.375 as a fraction.

(b)

Write

(43)

10.

62§%

as a decimal.

Find the tax on a $56.40 purchase

(46)

11.

Round

53,714.54 to the nearest

(42)

(a)

thousandth.

(b)

thousand.

if

the sales-tax rate

is

8%.

328

Saxon Math 8/7

12.

Find each missing exponent:

1471

10

(a)

13. (34)

c 5

10

•

o 2

n

= 10 u

10

(b)

ABCDEF

3.1

all

CD. If CD is 3 cm, what perimeter of the figure?

15.

A

angles are right angles and AF = AB = BC. Segment BC is twice the length of

14. In figure

dnv.2)

-f

The point marked by the arrow represents what decimal number?

3.0

1191

10 4 = 10

8

Use

a

Then

compass

to

draw

inscribe a regular

is

6

a

B

the

J D

a circle with a radius of 1 inch.

hexagon in the

circle.

(a)

What

is

the diameter of the circle?

(b)

What

is

the perimeter of the regular hexagon?

Solve:

w

16. (39)

100

1

(39)

= 1.5

18(35)

17.

x

19. 9.8

1.5

16 24

3.6

= x + 8.9

(35)

Estimate each answer to the nearest whole number. Then

perform the calculation. 20. (30)

111

4- + 5- + 5

3

2

21.

6^ -

5-1

8

(23, 30)

Simplify: 22.

A /16

•

3 23. 3.6 x 10

25

(20)

(47)

1

24.

8^

(26)

3

x

1 1 3- x -

5

3

25. (26)

1

3^ 8

26. 26.7 + 3.45 + 0.036 + 12 + 8.7 (35)

1

"i"

6^ 4

Lesson 47

27. (19,37)

The

below

one triangle rotated into three differen t positions. Dimensions are in inches. figures

illustrate

13

15

14

(a)

What

is

the perimeter of the triangle?

(b)

What

is

the area of the triangle?

28. Simplify

and compare: 125

-r

10 2

(47)

29.

329

O

Arrange these numbers in order from

O- 12 ^

x 10 2

least to greatest:

(30)

2 1 3' 2'

30. In this figure find the

5

_7_

12

'

6

measure of

(40)

(a)

Za.

(b)

Zb.

(c)

Describe

130°

how

measure of Zc.

to

find

the 65 c

330

Saxon Math 8/7

LESSON

48

Fraction-Decimal-Percent Equivalents

WARM-UP -x

Facts Practice: +

Decimals (Test

-r

J)

Mental Math:

c

$35.00 - = -

e.

Estimate: 6| x 3|

g.

10

a.

b.

7 x

x

8,

+

12.75

-r

10

10 2

.

d.

1,

-2,-2

f

,

f.

+

2,

x 4,

-

| of 80 2,

r

9,

6,

+

1, {~,

-r

2,

Problem Solving:

The counting numbers 1 through 9 are arranged in three columns. Each column contains three numbers, and the sum of the numbers in each column is the same. Describe how to find the sum of the numbers in each column. Then find that sum.

NEW CONCEPT We may

describe part of a whole using a fraction, a decimal, or a percent.

| of the circle is shaded. 0.5 of the circle is shaded.

50%

of the circle

is

shaded.

Recall that when we rename a fraction, we multiply by a form of 1 such as §, |, or Another form of 1 is 100%, so to .

convert a fraction or a decimal to a percent,

we

multiply the

number by 100%. Example

1

Solution

Write

^ as a percent.

To change a number the number by 100%. 210

x

to its percent equivalent,

100% =

^0% 10

= 70 o/o

we

multiply

Lesson 48

Example

2

Solution

331

Write § as a percent.

We

multiply by 100 percent.

?

x

ioo%

=

=

3

66-% 3

3

Notice the mixed-number form of the percent.

Example

3

Solution

Write 0.8 as a percent.

We

multiply 0.8 by 100%. 0.8 x

Example 4

Complete the

table.

100% = 80%

Fraction

Decimal

i

(a)

3

(e)

Solution

For

and

(a)

(b)

we

(b)

1.5

(c)

Percent

(d)

60%

(f)

find decimal

and percent equivalents of |.

0.3 (a)

3)1.00

(b)

|

100%

x

= 33 io/o

=

For (c) and (d) we find a fraction percent equivalent to 1.5. (c)

1.5

= 1

A = l! 10

and

For

(e)

(e)

60%

=

(f)

(or a

1.5

(d)

mixed number) and

100%

x

=

150%

2

we

find fraction

60 100

and decimal equivalents of 60%. =

60 = 0.6 100

Decimal

Percent

3

60%

(f)

5

ESSON PRACTICE Practice set*

Complete the

a

table.

Fraction 2

3

1.1

C.

e.

b.

a.

f.

d.

4%

332

Saxon Math 8/7

MIXED PRACTICE Problem set

1. (46>

2. l24 >

3. {u]

4. 1361

5. 1461

Ling pedaled hard. She traveled 80 kilometers in 2.5 hours. What was her average speed in kilometers per hour?

Write the prime factorization of 1008 and 1323. Then reduce gjjf.

1803 the United States purchased the Louisiana Territory from France for $15 million. In 1867 the United States purchased Alaska from Russia for $7.2 million. The purchase of Alaska occurred how many years after the purchase of the Louisiana Territory? In

Red and blue marbles were the marbles were red. (a)

What

(b)

What was

6.

7. (22)

fraction of the marbles

were blue?

the ratio of red marbles to blue marbles?

6-ounce can of peaches sells for 90$. A 9-ounce can of peaches sells for $1.26. Find the unit price for each size.

A

Which

(28>

in the bag. Five twelfths of

size is the better

buy?

The average of two numbers is the number halfway between the two numbers. What number is halfway between two thousand, five hundred fifty and two thousand, nine hundred? Diagram

this

statement.

Then answer

the questions

that follow.

Van has read five eighths of the 336-page

8. (48)

novel.

(a)

How many pages

has

(b)

How many more

pages does he have to read?

Complete the

table.

Van read?

Fraction i

2

Decimal

(e)

(b)

(a)

0.1

(c)

(f)

Percent

(d)

25%

333

Lesson 48

9.

m

The graph shows how one family spends income. Use this graph to answer (a)-(c). How Income

(a)

What percent

Is

their

annual

Spent

of the family's

income

is

spent on

fraction of the family's

income

is

spent on

"other"? (b)

What food?

(c)

If

$3200

is

spent on insurance,

how much

is

spent on

taxes? 10. Write 0.54 as a

decimal rounded

to three

decimal places.

(42)

11. (a) Estimate the length of

AB in

centimeters.

(32)

B (b)

Use a centimeter scale

to find the length of

AB to

the

nearest centimeter. 12. (a) Identify the

exponent and the base in the expression

5

3 .

(20, 47)

(b)

Find the missing exponent: 10 4

13. If the perimeter of a regular (18, 19)

14. (19)

how many Copy

hexagon

•

10 4 = icf—

is 1 foot,

each side

is

inches long?

this figure

on your paper.

Find the length of the unmarked

and find the perimeter of the polygon. Dimensions are in

12

sides,

centimeters. All angles are right

14

24

angles. 15. (46)

traveled 78 miles on 1.2 gallons of gas. averaged how many miles per gallon?

The moped

moped

The

334

Saxon Math 8/7

Solve: 16. ,39)

A

15

w

100

18. 1.44

17. (39)

6m

=

19

-

(30)

(35)

20

15 12

X

+ f = t \ O *

Simplify: 20. 2

5

+

l

4

+ 3

:

2

21.

V10

23.

8^ +

2

•

6

(20)

22. (30)

24 f 26 ;

26.

3^ 6

15

1— + ill 4

,

.

24

(23, 30)

1

25.

.

25

16

6j

9

f26j

2

4

1- 3

I

v

2^ - 4 3

Find the value of £ when a = $13.93 and b =

0.07.

(41, 45)

27. 3,

37)

The coordinates anci

( 5>

(

5j

of three vertices of a triangle are (-1, -1),

_ 4 ). what

the area of the triangle?

is

were asked how many siblings they had, and the answers were tallied. If one student from the class is selected at random, what is the probability that the selected student would have more than one sibling?

28. Students in the class 6,38)

Number

Number

of Siblings

of Students

mi

1

2

3

4 or more

29.

What

30.

Find the measures of Za,

m

(40}

the total price of a $50.00 item including 7.5% sales tax?

Zc

is

Z.b,

and

in the figure at right.

Q.

.

Lesson 49

E S S

335

O N

49

Adding Mixed Measures

/ARM-UP —

Facts Practice: +

x

-r

Decimals (Test

J)

Mental Math: a.

8 x S6.50

_

4

40

b.

25.75 x 10

d.

Estimate: 12.11

e.

x — 100 ',400

g.

Find the sum. difference, product, and quotient of

L

'

1.9

^ of 200

f.

|

and

|.

Problem Solving:

The teacher asked for two volunteers, and Adam. Blanca, and Chad raised their hands. From these three students, list the possible combinations of two students the teacher could select.

EW CONCEPT A

mixed measure is a measurement that includes different units from the same category (length, volume, time, etc.). Ivan

is

5 feet 8 inches

The movie was

tall.

hour 48 minutes

1

long.

To add mixed measures, we align the numbers in order to add units that are the same. Then we simplify when possible. Example

1

Solution

Add and

We

add

simplify: 1

like units,

yd

2

+ 2 3

change 15

in. to 1

ft

5

ft

to 1

yd 2 yd 2 yd 4

3 in. 3

Then we change

+ 2 yd 2

ft

8 in.

and then we simplify from 1

We

7 in.

ft

yd

yd

2

ft

7 in.

ft

8 in.

ft

15

ft

ft

4 vd 2

to 4

ft.

Now we have

3 in.

and add

ft

left.

in.

and add 5

right to

3 in.

to 3 yd.

Now we have

336

Saxon Math 8/7

Example

2

Add and

simplify:

+ Solution

We

add.

40 min 35 hr 45 min 50

2 hr

s

1

s

Then we simplify from +

s to 1

left.

2 hr

s

1

40 min 35 hr 45 min 50

s

min 85

s

3 hr 85

We change 85

right to

min 25 3 hr

Then we simplify 86 min

s

and add

86 min 25

to 1 hr

4 hr 26

to

85 min.

Now we have

s

26 min and combine hours.

min 25

s

LESSON PRACTICE Practice set*

a.

Change 70 inches

b.

Change 6

c.

Simplify: 5

d.

Add:

2

e.

Add:

5 hr

What

is

feet 3

yd

1

ft

and inches.

inches to inches.

20

ft

to feet

in.

8 in. + 1

42 min 53

s

yd

2

ft

9

in.

+ 6 hr 17 min 27

s

MIXED PRACTICE Problem set

1. (35 ' 45}

2. (44)

3. (46)

the quotient

when

the

sum

of 0.2

5.

is

Darren carried the football 20 times and gained a total of 184 yards. What was the average number of yards he gained on each carry? Write the answer as a decimal number.

Artemis bought two dozen arrows for six dollars. What was the cost of each arrow?

counted the sides on three octagons, two hexagons, a pentagon, and two quadrilaterals. Altogether, how many sides did he count?

4. Jeffrey

m

and 0.05

divided by the product of 0.2 and 0.05?

What

is

the

mean

of these

numbers?

(28, 35)

6.21, 4.38, 7.5, 6.3, 5.91, 8.04

.

337

Lesson 49

6. (22, 36)

Diagram

statement.

this

Then answer

the questions

that follow.

Only two ninths of the 72 The rest were cordial.

billy goats

(a)

How many of the billy goats

(b)

What was

were

gruff.

were cordial?

the ratio of gruff billy goats to cordial billy

goats? 7.

Arrange these numbers in order from

_

(42)

least to greatest:

.

0.5, 0.5, 0.54 8.

(a)

Estimate the length of segment

AB in inches.

(8)

B (b)

9.

Measure the length of segment eighth of an inch.

AB

to the nearest

Write each of these numbers as a percent:

(48)

(a)

10.

0.9

(b)

Complete the

table.

(c)

If

Fraction

(48)

I

Decimal

Percent

(a)

(b)

75%

(c)

(d)

5%

11. Mathea's resting heart rate is 62 beats per minute. (13)

she

is resting,

about

how many

While

times will her heart beat

in an hour? 12. (36)

13. (37)

What

the probability of rolling an even prime with one roll of a die (dot cube)? is

A

|-by-f-inch square a 1-by-l-inch square.

was

cut from

1

What was

the

in.

,

2

(a)

number

area

of

,r

the

original square? (b)

What that

(c)

14. (19)

is

the area of the square

was removed?

What

What

is

is

the area of the remaining figure?

the perimeter of the figure in problem 13?

1

in.

338

Saxon Math 8/7

15. 1371

The figures below show a triangle with sides 6 cm, 8 cm, and 10 cm long in three orientations. What is the height of the triangle

6

(a)

when

the base

cm?

8

(b)

is

cm?

Solve: i6. 09}

-10 (30;

-y- = ±» 100

17

45

09)

1

1

£

O

19.

1A

35 =

m

40

9d =

2.61

(35)

Simplify: 20.

VlOO + 4

3

21. 3.14 x 10 (47)

(20)

(

22.

3— + |4— — 2—

f23, J0J

2

24.

m

4^6

23.

days 8 hr 15 min + 2 days 15 hr 45 min

25.

3

26. $18.00

m

3- ^ 1-

6^

4

1

+ 2

yd yd

2

2

ft

6 in.

1

ft

9 in.

^ 0.06

(45J

27. Describe (29, 33)

28. l46)

how

to estimate the quotient

is

bat cost $18.50. The ball cost $3.50. What was the total price of the bat and ball including 6% sales tax?

The

LWH

if

L =

0.5,

W

=

0.2,

(41)

m

35.675

divided by 2|.

29. Evaluate:

30.

when

This quadrilateral

is

a rectangle.

Find the measures of Za, and Ac.

Z.b,

and

H

= 0.1

339

Lesson 50

LESSON

50

Unit Multipliers and Unit Conversion

WARM-UP —

—

'

Facts Practice: Proportions (Test I)

Mental Math: a.

5

C * 20

x

~

S48.00 100

e.

Estimate: 4| x if

g.

Start

7,

2 0.0125 x 10

d

',225

f.

with a half dozen, +

t8,-x

b.

+ 1,t

10,

-

4,

1

f

4Q

sq uare that number. -

2.

-

6.

10.

Problem Solving:

Copy

this

problem and

in the mis

fill

X

1001

NEW CONCEPT moment to review the procedure for reducing a When we reduce a fraction, we can replace factors

Let's take a fraction.

that appear in both the

numerator and denominator with

since each pair reduces to

l's.

1.

24 36

Also, recall that we can reduce before sometimes called canceling.

2

we

multiply. This

is

3 '

5

3 l

We

can applv

units before

this

we

procedure

to units as well.

multiply. 5

12

ft

1

'

1

in. ft

= 60

in.

We may

cancel

:

340

Saxon Math 8/7

we performed the division 5 ft -r X ft, which means, "How many feet are in 5 feet?" The answer is simply 5. Then we multiplied 5 by 12 in. In this instance

We remember

multiplying by a fraction

change the name of

3 to

^ q 6

The

^

fraction

is

the

of a

by multiplying by f

'

4 _ 12 " 4 4

name

another

name

number by whose value equals 1. Here we

we change

that

for 3

because 12 t 4 =

3.

When

the numerator and denominator of a fraction are equal (and are not zero), the fraction equals 1. There is an unlimited number of fractions that equal 1. A fraction equal to 1 may

have units, such as 12 inches

12 inches

Since 12 inches equals fractions that equal 1.

1

we can

foot,

12 inches

1

write two more

foot

12 inches

1 foot

Because these fractions have units and are equal to 1, we call them unit multipliers. Unit multipliers are very useful for converting from one unit of measure to another. For instance, if we want to convert 5 feet to inches, we can multiply 5 feet

by a multiplier feet units

that has inches

cancel and the product _

5

12

r/

n

-

„

1

Note that

on top and

5

ft

and

in. jv ft

on bottom. The

60 inches.

is

= cn 60

are equivalent.

5ft =

feet

.

in.

You may use

either form.

2£ 1

we want

96 inches to feet, we can multiply 96 inches by a unit multiplier that has a numerator of feet and a denominator of inches. The inch units cancel and the product is 8 feet. If

to convert

96*:.^

= 8ft

Lesson 50

341

we

selected a unit multiplier that canceled the wanted to remove and kept the unit we wanted in

Notice that

unit we the answer.

When we numbers

set

up unit conversion problems, we

in this order:

Given

1

Converted

measure

Write two unit multipliers for these equivalent measures: 3

Solution

Unit multiplier

X

measure

Example

will write the

We

=

ft

1

yd

write one measure as the numerator and

its

equivalent as

the denominator. 3

ft

lyd Example 2

Solution

lyd

and

3 ft

Use one of the unit multipliers from example (a)

240 yards to

(b)

240

feet to yards.

(a)

We

are given a

feet.

1 to

convert

feet.

measure in yards.

We

want the answer

in

So we write the following:

Unit

240 yd

ft multiplier

We want to

cancel the unit "yd" and keep the unit "ft," so we select the unit multiplier that has a numerator of ft and a denominator of yd. Then we multiply and cancel units. 3

240 yd

ft

lyd

= 720

ft

We know

our answer is reasonable because feet are shorter units than yards, and therefore it takes more feet than yards to measure the same distance. (b)

We are given the measure in feet, and we want the answer in yards. We choose the unit multiplier that has a numerator of yd. 240

ft

•

li£

= 80 yd

342

Saxon Math 8/7

our answer is reasonable because yards are longer units than feet, and therefore it takes fewer yards than feet to measure the same distance.

We know

Example

3

Solution

Convert 350 millimeters

to centimeters (1

cm

= 10 mm).

given millimeters and are asked to convert to centimeters. We form a unit multiplier that has a numerator

We

are

of cm.

350 nutf

1

Cm

•

10

= 35

cm

rrrnl

LESSON PRACTICE Practice set*

Write two measures:

multipliers

unit

yd = 36

a.

1

b.

100

c.

16 oz =

cm

each pair of equivalent

in.

=

1

m

1 lb

Use unit multipliers

to

d.

Convert 10 yards

e.

Twenty-four

f.

for

answer problems

d-f.

to inches.

feet is

how many

yards

(1

yd =

England 12 pence equaled

1 shilling.

24 shillings. This was the same as

how many

In old

3 ft)?

Merlin had pence?

MIXED PRACTICE Problem set

1. 135)

2. 136)

When

and

the product of 3.5

sum

of 3.5

The

face of the spinner

and

0.4,

what

0.4 is subtracted

is

the difference?

is

divided

from the

into ten equal parts. (a)

What

fraction of this circle

marked with (b)

is

a 1?

What percent marked with

of this circle a

number

is

greater

than 1? (c)

If

the spinner

will stop

is

spun, what

on a number

the probability that greater than 2? is

it

343

Lesson 50

3. (46)

The 13-ounce box of cooked cereal costs $1.17, while the 18-oimce box costs $1.44. Find the unit price for each

Which

size.

4. 1461

5. 1281

6. (13)

7. {22)

size is the better

buy?

Nelson covered the first 20 miles in 2| hours. What was his average speed in miles per hour?

The parking

lot

charges $2 for the

first

hour plus 50$

each additional half hour or part thereof. What is the charge for parking in the lot for 3 hours 20 minutes?

The

train traveled at

hour.

How long

Diagram

did

it

total

an average speed of 60 miles per take the train to travel 420 miles?

statement.

this

for

Then answer

the questions

that follow.

Forty percent of the 30 football players were

endomorphic. (a)

How many

(b)

What percent

of the football players

of

the

football

were endomorphic? players

were

not

endomorphic? 8. 181

9.

m 10.

Which percent

best identifies the

shaded part of this

circle?

A.

25%

B.

40%

C.

50%

D.

60%

Write 3| as a decimal number rounded to four decimal places.

Use exponents

to write 7.5 million in

expanded notation.

(47)

11. Write each

number

as a percent:

(48)

(a)

12. I4S)

0.6

Complete the

(b)

table.

±

Fraction

(c)

Decimal

H Percent

(a)

(b)

30%

(c)

(d)

250%

344

Saxon Math 8/7 13. List the

prime numbers between 90 and 100.

(21)

14.

this figure into a rectangle

The dashes divide

and a triangle.

(37)

12

8

cm

cm cm

18

15. dnv. 2)

(a)

What

is

the area of the rectangle?

(b)

What

is

the area of the triangle?

(c)

What

is

combined area of the rectangle and

the

Use a compass jYien use

a

measures

60°.

to

draw

triangle?

a circle with a radius of l| in.

draw

protractor to

Shade the sector

that

angle that formed by the 60°

central

a is

central angle.

Solve: 16. (39)

±° =

-Z-

X

18. 3.56

—=—

17.

42

= 5.6 - y

— 20

19.

(35)

4

1

(39)

(30)

=

w

+

— 15 y

20. t2,41}

Which property

is

illustrated

by each of the following

equations?

= xy + xz

(a)

x(y +

(b)

x + y = y + x

(c)

lx = x

z)

/

10 — 10

6

21.

Which

is

(47)

A. 10 22. (Inv 3)

3

equivalent to

^

B.

10 4

The coordinates of (0,-2), and (-2,0).

T 2

?

C.

1000

D. 30

three vertices of a square are

(2, 0),

(a)

What

(b)

Counting whole square units and half square units,

are the coordinates of the fourth vertex?

find the area of the square.

Lesson 50

by 4 children, how many

23. If 10 cookies are shared equally (44)

24. (Inv 4)

cookies will each child receive?

Below

a box-and-whisker plot of test scores. Refer to

is

the plot to answer (a)-(c).

4

h

10

5

20

15

(a)

What

is

the range of scores?

(b)

What

is

the

(c)

Write another question that can be answered by

median score?

Then answer

referring to the plot.

two

25. Write 1501

3 45

unit

multipliers

the question. the

for

conversion

mm

= 1 cm. Then use one of the unit multipliers to 10 convert 160 to centimeters.

mm

26. 4

yd

2

ft

+ 3 yd 5

7 in.

in.

(49)

\

27. (26,30)

5- 6

29. In (40)

1V

the

-r

2-

28.

3

(26, 30)

4 figure t

congruent

^ to

at

right,

ADCB.

3- +

AABC Find

3

8

7

is

the

measure of

30.

m

(a)

ABAC.

(b)

ABCA.

(c)

ZCBD.

Show two ways and

c =

3.

to evaluate a(b

-

c)

for a

=

4,

b =

5,

346

Saxon Math 8/7

Focus on

Creating Graphs Recall from Investigation 4 that we considered a stem-andleaf plot that a counselor created to display student test scores. If we rotate that plot 90°, the display resembles a vertical bar graph, or histogram. CO

CO

CO

in

CD

CO

CD

CO

LO

CM

CO in

CO

CD CD

OJ

LO

CM

00

CM

O o

o o cm

A

histogram

is

co

lo

a special type of bar graph that displays data

There are no spaces between the bars. The height of the bars in this histogram show the number of test scores in each interval. in equal-sized intervals.

Scores on Test 12

10

Z c

8

CD

§

6

CD

0-9

10-19

20-29

30-39

40-49

50-59

Score

1.

Changing the intervals can change the appearance of

a

histogram. Create a new histogram for the test scores itemized in the stem-and-leaf plot using the following intervals: 21-28, 29-36, 37-44, 45-52, and 53-60. Draw a break in the horizontal scale between and 21.

I

— Investigation 5

347

Histograms and other bar graphs are useful for showing comparisons, but sometimes the visual effect can be misleading. When viewing a graph, it is important to carefully note the scale. Compare these two bar graphs that display the same information. Car Sales

Car Sales 600

— 600

w

CO Cfl

400

V)

500

'c

§

D

200

400 Last

This

Last

This

Year

Year

Year

Year

2.

Which

3.

Larry made the bar graph below that compares his test score to Moe's test score. Create another bar graph that shows the same information in a less misleading way.

of the two graphs visually exaggerates the growth in sales from one year to the next? How was the exaggerated visual effect created?

Test Results

Larry's

Score

Moe's Score

K/" 0% 40%

50%

Changes over time are often displayed by line graphs. A double-line graph may compare two performances over time. The graph below illustrates the differences in the growing value of a S1000 investment compounded at 7% and at 10% annual interest rates. Compounded Value

of

S1000

at

7% and 10%

Interest

348

Saxon Math 8/7

4.

Create a double-line graph using the information in the table below. Label the axes; then select and number the scales. Make a legend (or key) so that the reader can distinguish between the

two graphed

Stock Values First

($)

XYZ

ZYX

Corp

Corp

1993

30

30

1994

36

28

1995

34

36

1996

46

40

1997

50

46

1998

50

42

Trade

lines.

Of

A

graph (or pie graph) is commonly used to show components of a budget. The entire circle, 100%, may represent monthly income. The sectors of the circle show how the income is allocated.

We

circle

see that the sector labeled "food"

circle,

we

representing

20%

20%

is

of the income.

To make

could draw a central angle that measures

20%

of the area of the a

20%

20%

sector,

of 360°.

of 360°

0.2 x 360° = 72°

With

we can draw 20% of a circle.

a protractor

a sector that

is

a central angle of 72° to

make

Investigation 5

5.

Create a pie graph for the table below to show how Kerry spends a school day. First calculate the number of degrees in the central angle for each sector of the pie graph. Next use a compass to draw a circle with a radius of about 2| inches. Then, with a protractor and straightedge, divide the circle into sectors of the correct size

and

label each sector.

How

Extensions

349

Kerry Spends a Day

Activity

% of Day

School

25%

Recreation

10%

Traveling

5%

Homework

10%

Eating

5%

Sleeping

40%

Other

5%

a.

Create a histogram for scores on a recent class

b.

Create a circle graph showing the percentages of students in the class with various eye colors.

c.

Explore

the

graph-creating

computer programs.

capabilities

test.

of

database

350

Saxon Math 8/7

MpB

Scientific Notation for

Large Numbers WARM-UP Facts Practice:

-f

-

x

4-

Decimals (Test

J)

Mental Math: a.

4 x $3.50

b.

2 4.5 x 10

n

-—r

= — x

d.

Convert 5

**•

20

2

-

2

km to

m.

| of 45

e.

15

g.

Find the sum, difference, product, and quotient of | and

5

f.

|.

Problem Solving: Beginning with 25.

the

1,

What number

is

first five

perfect squares are

1, 4, 9, 16,

and

the 1000th perfect square?

NEW CONCEPT The numbers used large or very small

in scientific

measurement

and occupy many places when written

standard form. For example, a light-year

9,461,000,000,000

way of decimal number and

Scientific notation

product of a

are often very

notation a light-year

is

a

is

in

about

km

expressing numbers as a a

power

of 10. In scientific

is 12 9.461 x 10

km

below we use scientific notation to approximate some common distances. Measurements are in millimeters.

In the table

Scientific notation

2

mm

width of pencil lead

mm

24

mm

diameter of a quarter

mm

160

mm

length of a dollar

3

mm

4500

mm

length of average car

4

mm

29,000

mm

length of basketball court

mm

110,000

mm

length of football field

mm

1,600,000

mm

one mile

mm

42,000,000

mm

distance of runner's marathon

1

2.4 x 10

10

4.5 x 10 2.9 x 10 x

10

1.6 x

10

4.2

10

1.1


2.1 x 10

1

352

Saxon Math 8/7 Scientific calculators will display the results of

an operatior

in scientific notation if the number would otherwise exceec the display capabilities of the calculator. For example, tc

mu ltiply

one million by one million,

we would

enter

contains more digits than can be displayed by many calculators. Instead of displaying one trillion in standard form, the calculator displays one trillion in some modified form of scientific notation such as

The answer, one

trillion,

"-

t

(

or perhaps

I.

x

in'*

\u

LESSON PRACTICE Practice set

Write each number in scientific notation: a.

15,000,000

b.

400,000,000,000

c,

5,090,000

d.

two hundred

fifty billion

Write each number in standard form: e.

3.4 x 10*

5 x 10*

f

.

g.

1

x 10'

Compare: h.

i.

O one million Q 1.5

x 10

5

1.5 1

x

10 6

x 10

{

MIXED PRACTICE Problem set

Refer to the double-line graph 1

and

below

to

answer problems

2.

Test Scores 100

90

4

"

80

'

——

"

1

_ _ u

y"'-

^^^^^

8 70 Bob': 5 score

60

Clasi 5 average

50

3 Test

1

1. (38)

2. (28, 38)

On how many

tests

was Bob's score

better than the class

average f

What was Bob's average

score on these five tests?

353

Lesson 51

3. In ll9)

the pattern on a soccer ball, a

and

hexagon

regular

pentagon share a

a

common

regular side. If

the perimeter of the hexagon

is

what

of

the

is

perimeter

9

in.,

the

pentagon? 4. (46>

The

40$ per can or 6 cans for $1.98. can be saved per can by buying 6 cans at the

store sold juice for

How much

6-can price? 5. ll4, 36)

Five sevenths of the people who saw the phenomenon were convinced. The rest were unconvinced. (a)

What

fraction of those

who saw

the

phenomenon

were unconvinced? (b)

What was

the

ratio

of

the

convinced

to

unconvinced? 6.

(a)

Write twelve million in scientific notation.

(b)

Write 17,600 in scientific notation.

(a)

Write 1.2 x 10 4 in standard form.

(b)

Write 5 x 10 6 in standard form.

(51)

7. (51)

8.

Write each number as a decimal:

(43)

(a)

9.

(b)

I

Round each number

87^%

to the nearest

thousand:

(33)

(a)

10. (48)

11. (Inv,5)

29,647

Complete the

(b)

table.

Fraction

5280.08

Percent

Decimal

(a)

(b)

40%

(c)

(d)

4%

Find the number of degrees in the central angle of each sector of the circle shown.

12£% 25% (b)

(c)

(d)

the

354

Saxon Math 8/7

12.

m

What

the total price, including

is

5%

sales tax, of

$15.80 item?

13. Layla is thinking of a positive, single-digit, (36)

Lou guesses guess

14. 1181

a

it

is

7.

What

is

even number.

the probability that Lou's

correct?

is

These two quadrilaterals are congruent. Refer figures to answer (a) and (b).

to these

r

WXYZ is

(a)

Which

(b)

Which segment

angle in

in

congruent to

ABCD

is

AA

in

congruent to

AB CD?

WX

in

WXYZ? Refer to the figure below to answer problems 15 and 16. Dimensions are in meters. All angles are right angles.

14

18

15.

What

is

the perimeter of the figure?

What

is

the area of the figure?

(J 9)

16. (37)

Solve:

(39)

Mx

19.

5/77

17

(35)

60 40

= 8.4

n

18(39)

4.2

20. 6.5 (35)

7

- y = 5.06

Lesson 51

355

Simplify: 21. 5

2

+ 3 3 + V64

22. 16

(49)

days 18 hr 50 min + 2 days 8 hr 25 min 5

(

25. (23, 30)

27. t41>

10 1

6- + 3

1 7 5- - 3^

\

4

m

3

+

1

yd yd

2

ft

cm 5 in.

9 in.

\

26. 3^

8j

Show two ways and y =

24.

mm

•

(50)

(20)

23.

cm

(26)

to

3

3

evaluate x(x + y)

2

for

x =

0.5

0.6.

The coordinates of three vertices of a triangle are A (-4, 0), B (0, -4), and C (-8, -4). Graph the triangle and refer to it to answer problems 28 and 29. 28. Use a protractor to find the measures of ZA, ZB, and ZC. (17)

29.

What

is

the area of A ABC?

(37)

30. 1321

When

temperature increases from the freezing temperature of water to the boiling temperature of water, it is an increase of 100 degrees on the Celsius scale. The same increase in temperature is how many degrees on the Fahrenheit scale? the

356

Saxon Math 8/7

LESSON Order of Operations Mr

Facts Practice: Powers and Roots (Test K)

Mental Math: a.

6 x 750

c.

^

= 3

f - 20

e.

10

g.

At 80

km

4.5

d.

Convert 250

2 f.

per hour,

how

10 2

b.

cm to m.

^ of 200

far will a car travel in

l\ hours?

Problem Solving: Find x

if

3x +

5 = 80. Explain

your thinking.

NEW CONCEPT Recall that the four fundamental operations of arithmetic are addition, subtraction, multiplication, and division. We can also raise

numbers

to

powers or find

their roots.

When more

than one operation occurs in the same expression, perform the operations in the order listed below.

we

Order of Operations 1.

Simplify powers and roots.

2.

Multiply and divide in order from

3.

Add and

Note:

If

left to right.

subtract in order from left to right.

there are parentheses (or other enclosures),

we

simplify within the parentheses, from innermost to outermost, before simplifying outside the parentheses.

The

each word in the sentence "Please excuse my dear Aunt Sally" reminds us of the order of operations: parentheses (or other symbols of inclusion), exponents (and initial letter of

roots), multiplication

and division, addition and subtraction.

Lesson 52

Example

1

Solution

Simplify: 2 +

We

left to right

before

we

subtract. 2

4x3-4

+

2

Solution

+ 2

multiply and divide in order from

add or

Example 2

4x3-4

3 57

Simplify:

3

2

+

+

problem

+ 2

12-2

multiplied and divided

12

added and subtracted

3

5

•

A

division bar can serve as a symbol of inclusion, like parentheses. We simplify above and below the bar before dividing. 3

2

+

3

5

•

problem

2

3-5

9 +

simplified

power

2

9 + 15

multiplied above

24

added above

2

divided

12

Example

3

Solution

Evaluate: a + ab

We

if

a = 3

and b = 4

begin by writing parentheses in place of each variable. This step may seem unnecessary, but many errors can be avoided if this is always our first step. will

a + ab

() + ()()

Then we

parentheses

replace a with 3 and b with

4.

a + ab (3)

We

+

(3)(4)

substituted

follow the order of operations, multiplying before adding. (3)

3

+

(3)(4)

problem

+ 12

multiplied

15

added

358

Saxon Math 8/7

Example 4 Solution

xy - |

Evaluate:

we

First

if

x =

9

2 and y = -

replace each variable with parentheses.

x

xy

2

()()-— 2

parenthes es

write 9 in place of x and § in place of y.

Then we

x

xy

2 (?)

(9)

substituted

2

We

follow the order of operations, multiplying and dividing before we subtract.

'2^

(9)

(9)

problem

2

«-4j

multiplied and divided

subtracted

GD CD CD CD

C3SSQ CD CD CD CD CD CD CD CD CD CD CD CD CD CD GD CD

Calculators with algebraic-logic circuitry are designed to perform calculations according to the order of operations.

perform circuitry without algebraic-logic calculations in sequence. You can test a calculator's design by selecting a problem such as that in example 1 and entering the numbers and operations from left to right, concluding with an equal sign. If the problem in example 1 is used, a displayed answer of 12 indicates an algebraic-logic design. Calculators

LESSON PRACTICE Practice set*

Simplify:

+

a.

5

b.

50 -

8-

c.

24 -

8-

2 d.

3

-

5

5

+ 3

2

5

-r

5

+

6^3

6-

2^4

5

+ 2

•

5

Lesson 52

359

Evaluate: e.

f.

ab - be

a =

if

ab + —

a =

if

b =

5,

b =

6,

3,

4,

and

and

c

= 4

c = 2

c

g.

VI

IX ED

x - xy

if

x =

|

and y = |

PRACTICE

Problem set

1. (21)

If

the product of the

by the sum of the

numbers is divided three prime numbers, what is the

first

first

three prime

quotient?

2. tl8)

3. (3i, 35)

Sean counted a total of 100 sides on the heptagons and nonagons. If there were 4 heptagons, how many nonagons were there? Twenty-five and two hundred seventeen thousandths is 10W muc ]1 } ess than two hundred two and two hundredths? j

4.

Jermaine bought a pack of

3

blank tapes

for $5.95.

46)

(a)

What was

(b)

The

the price per tape to the nearest cent?

sales-tax rate

was 7%. What was the

total cost of

the three tapes including tax?

5. 28)

Ginger is starting a 330-page book. Suppose she reads for 4 hours and averages 35 pages per hour. (a)

How many pages

(b)

After 4 hours,

will she read in 4 hours?

how many

pages will she

still

have

to

read to finish the book? 6. 22)

the following statement, convert the percent to a reduced fraction. Then diagram the statement and answer In

the questions. Seventy-five

disembarked (a)

How many

(b)

What percent

percent

of

the

60

at the terminal.

passengers disembarked

the terminal?

passengers

at

the terminal?

of the passengers did not disembark at

360

Saxon Math 8/7

7.

(a)

Write 3,750,000 in scientific notation.

(b)

Write eighty million in scientific notation.

(a)

6 Write 2.05 x 10 in standard form.

(b)

1 Write 4 x 10 in standard form.

(51)

8. (51)

9.

m

Write each number as a decimal: (a)

(b)

|

10. Write 3.27 as a l42)

11.

I

6.5%

decimal number rounded to the nearest

thousandth.

Complete the

table.

Decimal

Fraction

(48)

Percent

(a)

(b)

250%

(c)

(d)

25%

by 9 and write the answer

12. Divide 70 (44)

13. 1341

mixed number.

(a)

as a

(b)

as a decimal

number with

a bar over the repetend.

What decimal number names

the point

(32)

Draw

a rectangle that

answer

(a)

(a)

What

(b)

What

and

is

is

1.0

cm

3

long and 2

cm

wide. Then

(b).

the perimeter of the rectangle in millimeters?

is

the

area

of

the

rectangle

in

centimeters? 15. In l37)

the

arrow?

0.9

14.

marked by

quadrilateral

parallel to BC.

ABCD,

d 6

(a)

Find the area of A ABC.

(b)

Find the area of AACD.

(c)

What

the

is

Dimensions are in

centimeters.

is

AD

combined area of the two

triangles?

square

361

Lesson 52

Solve:

1C lb.

8 —

(39)

f

18.

56 105

=

p +

17 (39)

6.8 = 20

— 15

2.5

-

3.6

= 6.4

21. 4 +

4-

4-

19. g (35)

(35)

Simplify: 20. 5

3

- 10 2

\25

4 + 4

(20)

4 8 ~ '

22. f35.45j

°- 24

23.

m

(0.2) (0.6)

hr 45 + 2 hr 53 5

min 30 min 55

s

s

\

24. (26, 30)

6- + 4

5-

25.

3

sum

26. Estimate the (35}

(26. 30)

J

5- -

3- + 4

2

to the nearest

2

whole number. Then

perform the calculation. 8.575 + 12.625 + 8.4 + 70.4

27. 0.8 x 1.25 x

10

b

(47)

28. Evaluate:

ab + §

if

a = 4 and b =

0.5

f52j

29. Convert 1.4 meters to centimeters (1

m

= 100 cm).

(50)

30. (36)

The students favorite sport.

in a class of 30

Twelve said

were asked

to

name

their

football, 10 said basketball,

and 8 said baseball. If a student is selected at random, what is the probability that the student's favorite sport is basketball?

— 362

Saxon Math 8/7

LESSON Multiplying Rates 53 H£>Jgl WARM-UP Facts Practice: Powers and Roots (Test K)

Mental Math: a.

8 x $1.25

c.

2x +

e-

(If

g.

10 x

—

1,

= 75

5

b.

12.75 x 10

d.

Convert 35

cm to mm.

I of 45 3, double that number, + 1, f-

+ 4, aT, x 5, square that number

6, -r

x

8,

Problem Solving: Allen wanted to form a triangle out of straws that were 5 in., 7 in., and 12 in. long. He threaded a piece of string through the three straws, pulled the string tight, and tied it. What was the area of the figure?

NEW CONCEPT If

we

hour (50 mph), we we would travel in 4 hours by

are traveling in a car at 50 miles per

can calculate

how

far

multiplying.

50 mi —

iz

.

4 hr x

~

or> = 200 mi

1 fer

We

can find out

how

long

it

will take us to travel 300 miles

by dividing. onn mi 300 .

.

-f

— mi 50

-

1

300 ml

x

hr

-12* = 50 ml

300^

= 6hr

50

Notice that dividing by a rate is similar to dividing by a fraction. We actually multiply by the reciprocal of the original rate. There are two forms of every rate and they are reciprocals. Thus we may solve rate problems by multiplying

— 3 63

Lesson 53

by the correct form of the

The following statement

rate.

expresses a rate:

There were 5 chairs in each row.

We

can express

this rate in

5 chairs —

^

(

(a)

we can

rate (a)

we can

(b)

number

^ c hairs 1

Using rate 20 chairs.

x&W

—row — 1

of chairs in 6 rows.

_ go

c j ia i rs

number

find the

on cnalrs x 20

row

5 chairs

find the

6 rjawS x

1

ru ^ (b)

row

1

Using

two forms.

-

of rows

needed

for

- = 4„ rows

5 cnalrs

Example

1

Solution

Eight ounces of the solution costs 40 cents. (a)

Write two forms of the rate given by this statement.

(b)

Find the cost of 32 ounces of the solution.

(c)

How many

(a)

The two forms

ounces can be purchased of the rate are

40 cents

8 oz 40 cents (b)

To

for $1.20?

find the cost,

we

8 oz

—

no Qt x 40 cents 32 8 p£ .

money on

use the form that has i

top.

,

canceled ounces

1280 — -— cents

u multiplied

160 cents

simplified

.

,

.

We

usually write answers equal to a dollar or more by using a dollar sign. Thus the cost is $1.60. (c)

To

find the

ounces on

number

of ounces,

we

use the form that has

top.

120 cents x

8 oz

—

canceled cents

40 gents

960 — oz 40 -

u r multiplied r

24 oz

simplified

.

,

Saxon Math 8/7

Note that

can be reduced. Both forms of example 1 can be reduced.

rates, like fractions,

the rate in part

(a)

\

1

& oz

£6

oz

I

oz

2

Solution

Jennifer's

can be used

rates

we saw

reducing as

Example

to solve

in parts (b)

and

first

(c).

(a)

Write the two forms of the rate given by this statement.

(b)

How far

(c)

How long would

(a)

The two forms

did she drive in 5 hours? take her to drive 300 miles?

it

of the rate are

1 (b)

To

find

how

far,

5

we

(c)

To

find

1

hour

use the form with miles on top.

60

beofS x

paur

how much time, we use 300

mire's x

= 300 miles

the form with time on top.

^°VL

60 miles

pencils cost $0.25 each, buy for $2.00? If

We

hour

60 miles

1

Solution

problems without

speed was 60 miles per hour.

60 miles

Example 3

oz

1

5

However,

cents cents _ 5

4rQ

5 cents

cents"

can use rates to solve pencil has two forms.

how many

this

= 5 hours

pencils can Carol

find

how many, we $2.00 x

Ann

problem. The rate of $0.25 per

1 pencil $0.25 and 1 pencil $0.25

To

of

use the form with pencils on top.

= 8 pencils

Lesson 53

3 65

ESSON PRACTICE Practice set

In the lecture hall there

were 18 rows. Fifteen chairs were in

each row. a.

Write the two forms of the rate given by this statement.

b.

Find the

total

A car could travel

number

of chairs in the lecture hall.

24 miles on one gallon of gas.

c.

Write the two forms of the rate given by this statement.

d.

How many gallons would

it

take to travel 160 miles?

1IXED PRACTICE

Problem set

Refer to this double-bar graph to answer problems 1-3: Distribution of Birthdays in Class

1

1

mmm

Boys Girls

Jul.-Sept.

Oct.-Dec. ..

.

2

1.

(a)

(38)

(b)

2. (38)

3. (38>

4. (28 46} -

4

3

Number

of

5

Students

How many boys are in the class? How many girls are in the class?

What

percent of the students have a birthday in one of the months from January through June?

What

fraction of the boys

have a birthday in one of the

months from April through June? At the book fair Muhammad bought 4 books. One book cost $3.95. Another book cost $4.47. The other 2 books cost $4.95 each. (a)

What was

(b)

What was sales tax?

the average price per book? the total price of the books including

8%

366

Saxon Math 8/7

5. (22}

Diagram

Then answer

statement.

this

the questions

that follow.

Seven twelfths of the 840 gerbils were hiding in their burrows. (a)

What

were not hiding in

fraction of the gerbils

their

burrows?

6.

(b)

How many gerbils

(a)

Write one trillion in scientific notation.

(b)

Write 475,000 in scientific notation.

(a)

2 Write 7 x 10 in standard form,

(b)

Compare:

were not hiding in their burrows?

(51)

7. (51)

8.

m

9. {27]

2.5 x 10

Use unit multipliers (a)

35 yards to feet

(b)

2000

cm

to

Use prime

m

to

6

Q

5

perform the following conversions: =

1

cm

=

(3 ft

(100

2.5 x 10

factorization

yd) 1

m) find

to

the

least

common

multiple of 54 and 36.

10. Estimate the difference of 19,827

and 12,092 by rounding

(29>

to the nearest

thousand before subtracting.

11.

Complete the

table.

12. Write

Decimal

Fraction

(48)

Percent

(a)

(b)

150%

(c)

(d)

15%

each number as a percent:

(48)

(a)

(b)

|

how many

m

Stephanie is 165 cm tall. Big centimeters taller than Stephanie?

13. Big Bill is 2 (32}

tall.

14. Refer to this figure to (19,37)

0.06

(a)

and

feet.

All

answer

Dimensions are in angles are right angles. (a)

What

(b)

What

is

figure?

is

the area of the figure? the perimeter of the

12

Bill is

367

Lesson 53

15.

The bank would exchange

(53)

1

1.6

Canadian dollars (C$)

for

U.S. dollar (US$).

(a)

Write two forms of the exchange

(b)

How many exchange

for

rate.

would

Canadian dollars 160 U.S. dollars?

the

bank

Solve: 16. (39)

18 100

6 = 1.5 9

17

P

(39)

= 7.25 +

18. 8

1

90

m

Wn

=

19. 1.5

(35)

(35)

Simplify: 20.

\81 + 9 2 -

2

5

21. 16

(20)

t 4 t 2 + 3 x 4

(52)

(

22.

3

yd

1 ft

7\

in.

2

6|

in.

23.

(49)

(26. 30)

+

24. (26, 30)

12- +

8- -

ft

1-

5

•

\

5-

-f

2-

3

3-

2

5

25. 10.6 + 4.2 + 16.4 + (3.875 x 10 1 ) (35, 47)

26. Estimate: 6.85 x

4-^ 16

(29, 33)

27. Evaluate: (52)

28. Petersen (44)

30. (40)

if

needed

2\ dozen eggs.

a =

to

How

6,

b = 0.9, and c = 5

pack 1000 eggs into many flats could he

flats that

held

fill?

one chance in five of guessing the correct answer, then what is the probability of not guessing the correct answer?

29. If there (36)

^ DC

is

Find the measures of angles

a, b,

and

c in this figure:

368

Saxon Math 8/7

LESSON

54

Ratio

Word Problems

WARM-UP Facts Practice: Fraction-Decimal-Percent Equivalents (Test L)

Mental Math: a.

4 x $4.50 12

c.

12.75

d.

Convert 1.5

3

V900 -

8-

Mentally perform each calculation:

f.

3,2

4

5

1_2 4

-r

1,2 4*5

5

m to cm.

^ of 90

e.

3

10

b.

1_;_2

4'5

Problem Solving: At

first

| of the students in the

\ of the remaining students the room?

room were

were

girls.

girls.

When

How many

boys left, girls were in 3

NEW CONCEPT In this lesson

we

will use proportions to solve ratio

problems. Consider the following ratio

The

ratio of parrots to

word problem:

macaws was

there were 750 parrots,

word

5 to

7.

If

how many macaws

were there?

problem there are two kinds of numbers, ratio numbers and actual count numbers. The ratio numbers are 5 and 7. The number 750 is an actual count of parrots. We will arrange these numbers into two columns and two rows to form a ratio box. Practicing the use of ratio boxes now will pay dividends in later lessons when we extend their application to more complex problems. In

this

Ratio

Actual Count

Parrots

5

750

Macaws

7

M

3 69

Lesson 54

We

were not given the actual count of macaws, so we have used to stand for the number of macaws. The numbers in this ratio box can be used to write a proportion. By solving the proportion, we find the actual count of macaws.

M

Ratio

Actual Count

Parrots

5

750

Macaws

7

M

— —

5

750

7

M

5M

= 5250

M= We Example

In the auditorium the ratio of boys to girls

were 200 Solution

macaws was

find that the actual count of

girls in

1050.

was

5 to 4. If there

how many boys were

the auditorium,

We begin by making

1050

there?

a ratio box.

Ratio

Actual Count

5

B

4

200

We

use the numbers in the ratio box to write a proportion. Then we solve the proportion and answer the question. Ratio

Actual Count

Boys

5

B

Girls

4

200

— —

5

B

4

200

4B = 1000 B = 250 There were 250 boys in the auditorium.

ESSON PRACTICE Practice set

Solve each of these ratio word problems. Begin by making a ratio box. a.

The

girl-boy ratio

many boys b.

was

9 to

7.

If

63

girls

attended,

how

attended?

sparrows to blue jays in the yard was 5 to 3. If there were 15 blue jays in the yard, how many sparrows were in the yard?

The

ratio of

370

Saxon Math 8/7

c.

untagged fish was 2 to 9 were tagged. How many fish were untagged?

ratio of tagged fish to

The

Ninety fish

d. Calculate the ratio of

Then

boys

to girls in

your classroom.

calculate the ratio of girls to boys.

MIXED PRACTICE Problem set

1. (12. 28)

2. l28)

3. (54>

4. (53)

5. l6 271 '

6. [22]

on the

anniversary of the signing of the Declaration of Independence. He was born in 1743. The Declaration of Independence was signed in 1776. How many years did Thomas Jefferson live?

Thomas

Jefferson died

The heights

fiftieth

of the five basketball players are 190 cm,

195 cm, 197 cm, 201 cm, and 203 cm. What is the average height of the players to the nearest centimeter?

Use a ratio box to solve this problem. The ratio of winners to losers was 5 to 4. If there were 1200 winners, how many losers were there?

What

the cost of 2.6

is

pounds of cheese

at

$1.75 per

pound?

What

is

and 6

is

common multiple of 4 divided by the greatest common factor of 4 and 6?

the quotient

Diagram

this

when

statement.

the least

Then answer

the questions

that follow.

Eighty percent of the 80 trees were infested.

7.

(a)

How many trees were

(b)

How many trees

(a)

Write 405,000 in scientific notation.

(b)

Write 0.04 x 10 5 in standard form.

infested?

were not infested?

(51)

8.

Find each missing exponent:

(47)

(a)

10

6 •

10 2 = 10

a

(b)

10 6

-r

10

2

= 10 a

Lesson 54

9.

Use unit multipliers

perform the following conversions:

to

(50)

(a)

5280

(b)

300

feet to

cm

to

yards

mm

(1

(3 ft

cm

=

yd)

1

= 10

mm)

3.1415926 as a decimal number rounded decimal places.

10. Write (33)

11. (blv 5)

Find the number of degrees in the central angle of each sector of the circle shown.

\(d)

20% (c)

\V

30%

(53)

13.

to four

/\10%

/

\

12.

371

(b)

A

\

\

A train is traveling at a steady speed of 60 miles per hour. (a) How far will the train travel in four hours? (b) How long will it take the train to travel 300 miles? Which

is

equivalent to

06 —

?

1

2

(20)

A.

2'

B.

Z

C.

r

D. 3

Refer to the figure below to answer problems 14 and 15, Dimensions are in centimeters. All angles are right angles. 5

14.

What

is

the perimeter of the figure?

What

is

the area of the figure?

(19)

15. (37)

16. (2.

17. (34 35> '

Name

each property

illustrated:

41)

= 1

(a)

\ +

(b)

5(6 + 7) = 30 + 35

(c)

(5

+

6)

+ 4 = 5 +

(6

+

4)

Draw a square with sides 0.5 inch long. (a) What is the perimeter of the square? (b)

What

is

the area of the square?

372

Saxon Math 8/7

18.

The box-and-whisker plot below was created from student scores (number of correct out of 20) on the last math test. Do you think that the mean (average) score on the test is likely to be above, at, or below the median score? Explain your answer.

20

15

10

Solve: 19. 6.2 =

x +

20.

4.1

(35)

(35)

21. (39)

?i = 36

22.

27

(35)

r

w

= 6.25

Simplify: 2

23. II

+

l

3

- Vl21

25 (35)

24. (52)

(20)

(2 5)2 -

26.

2(2.5)

(49)

+ 2 weeks 6 days 10 hr (

27. 3-^(23,30)

29. dnv.3)

10

+ (9^ - 6§^

28.

3J

(26)

^2

6,3-

O

The coordinates of the vertices of A ABC are A (-1, 3), 5( _ 4? 3)j and c( _ 4) _ a) The coorciinates of AXYZ are X(l, 3), 7(4, 3), and Z (4,-1). Graph A ABC and AXYZ.

30. Refer to the

m

7\

(a)

(b) (c)

graph drawn in problem 29 to answer

(a)-(c).

Are A ABC and

AXYZ similar? Are A ABC and AXYZ congruent? Which angle in A ABC corresponds

to

ZZ in AXYZ?

Lesson 55

.

E S S

3 73

O N

55

Average, Part 2

VARM-UP

Facts Practice: +

-

x + Decimals (Test

J)

Mental Math: a.

20 x $0.25

b.

0.375 x 10 2

c.

2x -

d.

Convert 3000

e.

g.

5

= 75

212 f.

| of

how many

At 30 pages an hour,

m to km.

100

pages can Mike read in 2|

hours?

Problem Solving:

Copy

this

problem and

fill

_3_

in the missing digits:

IEW CONCEPT If

the average of a group of numbers and how many are in the group, we can determine the sum of the

we know

numbers numbers.

Example

1

Solution

The average

We

of three

numbers

is 17.

What

is

their

sum?

what the numbers are, only their average. Each of these sets of three numbers has an average of 17: are not told

16 + 17 + 18 _ 51 _

"3

3

10 + 11 + 30

51

3

3

1

1+49

+ ~

3

= 17

374

Saxon Math 8/7

Notice that for each set the sum of the three numbers is 51; Since average tells us what the numbers would be if they were "equalized," the sum of the three numbers is the same as if each of the numbers were 17.

17 + 17 + 17 = 51 Thus, the

sum Example

2

Solution

number of numbers times

their average equals the

of the numbers.

The average of four numbers is 25. If three of the numbers 16, 26, and 30, what is the fourth number? If

the average of four

numbers

is

25, their

sum

is

are

100.

4 x 25 = 100

numbers. The sum of these three numbers plus the fourth number, n, must equal 100.

We

are given three of the

16 + 26 + 30 +

17

= 100

The sum of the first three numbers is 72. Since the sum four numbers must be 100, the fourth number is 28.

of the

16 + 26 + 30 + (28) = 100

Example 3

Annette need on her Solution

What score does test to bring her average up to 90?

After four tests Annette's average score fifth

was

89.

Although we do not know the specific scores on the first four tests, the total is the same as if each of those scores were 89.

Thus the

total after four tests is

4 x 89 = 356

The

her first four scores is 356. However, to have an average of 90 after five tests, she needs a five-test total of 450. total of

5 x

90 = 450

Therefore she needs to raise her total from 356 to 450 on the fifth test. To do this, she needs to score 94.

356 four-test total + 94 fifth test 450 five-test total

3 75

Lesson 55

ESSON PRACTICE Practice set

a.

Ralph scored an average of 18 points in each of his first five games. Altogether, how many points did Ralph score in his first five games?

b.

The average of four numbers is 45. If three of the numbers are 24, 36, and 52, what is the fourth number?

c.

After five tests, Tisha's average score tests,

her average score was 89.

was 91. After six What was her score on the

sixth test?

/IIXED

PRACTICE

Problem set

1. 1541

Use a

box

ratio

sailboats to rowboats in the

56 sailboats in the bay, 2. t55)

3. l46)

4.

The

solve this problem.

to

how

ratio

of

bay was 7 to 4. If there were many rowboats were there?

The average of four numbers is 85. If three of the numbers are 76, 78, and 81, what is the fourth number?

A

one-quart container of oil costs 89(2. A case of 12 onequart containers costs $8.64. How much is saved per container by buying the oil by the case?

BC is how much

Segment

AB?

longer than segment

(8)

C

B

A I

I

I

|

|

I

I

|

I

I

|

I

I

|

|

|

I

I

|

i

I

|

i

i

|

i

|

|

|

|

|

2

3

|

inch

5. (22)

1

Diagram

this

statement.

Then answer

4'

the questions

that follow.

Three tenths of the 30 students earned an A.

6.

(a)

How many students

(b)

What

(a)

Write 675 million in scientific notation.

(b)

5 Write 1.86 x 10 in standard form.

earned an A?

percent of the students earned an

A?

(51)

7. l47>

Find each missing exponent: 2 8 10 = 10^ (a) 10 •

(b)

10 8

4-

10 2 =

1

9. 1311

perform the following conversions:

to

(a)

24 feet to inches

(b)

500 millimeters

to centimeters

Use digits and other symbols to write "The product of two hundredths and twenty-five thousandths is five tenthousandths."

10. (46>

11.

What

the total price of a $3.25 drink including 7% sales tax? is

Complete the

table.

Decimal

Fraction

(48)

sandwich and a $1.10

i

(b)

(a)

5

0.1

(c)

(e)

(a)

Which segment

(d)

75%

(f)

D

Refer to the figure at right to answer problems 12 and 13. 12.

Percent

J

parallel to

is

BC?

[7]

(b)

Which

two

segments

perpendicular to (c)

ABC

Angle

is

1

are

r

B

H

BC?

an acute angle. Which angle

is

an

obtuse angle?

AD

13. If

= 6 cm,

CD

14. l28)

15. (53)

(a)

what

is

the area of rectangle

(b)

what

is

the area of triangle

(c)

what

is

the area of figure

Donato is

is

6 feet 2 inches

how many

CB =

= 8 cm, and

(37)

tall.

10 cm, then

AHCD?

ABH?

ABCD? Bob

is

68 inches

tall.

Donato

inches taller than Bob?

Monte swam 5 laps in 4 minutes. (a) Write two forms of this rate. (b)

How many

laps could

Monte swim

in 20 minutes at

this rate? (c)

How

long would

this rate?

it

take for

Monte

to

swim 20

laps

at

.

377

Lesson 55

16.

Show two wavs

to evaluate b(a

-

b)

for a

= \ and b =

\.

Solve:

„/

30 70

21

18,

1000

=

2.5

U"

A"

Estimate each answer

to

whole number. Then

the nearest

perform the calculation. 19.

2—

(30)

12

+ 6§ + 4^ 6

20.

-

6

( -1

8

5

3

;

Simplify 21. io

vd

vd

1

23. 12-

3

25. _-f

22.

•

- 4

:

-

2~-

5-

3

4

27. 3.47 -

- -.144

24.

50 - 30 -

26.

3- 3

.-.

- 1.359)

28.

(45$

"

quadrilateral

is

a

rectangle.

Find the measures of angles and c.

a.

b.

2

5

1

3

-i-

22

(0.6l(0.28)[0.01

29. SI. 50 - 0.075

4

7 in.

ft

5 in.

(35)

30. This

2

1

10 (6

yd

-

112—

6-

8

j

-

6

378

Saxon Math 8/7

LESSON

56

Subtracting

Mixed Measures

WARM-UP Facts Practice: Fraction-Decimal-Percent Equivalents (Test L)

Mental Math: a. c. e.

g.

10

30 x 2.5

b. 0.25

3x + 4 = 40 25 2 - 15 2 Square 9, - 1, f

d.

Convert 0.5

f.

^ of $50.00

2,

-

4, {~,

-r

x 3, + 2,

-r

5,

m to cm. f, -

5.

Problem Solving:

A

palindrome is a word or number that reads the same forward and backward, such as "mom" and "121." How many three-digit numbers are palindromes?

NEW CONCEPT have practiced adding mixed measures. In this lesson w will learn to subtract them. When subtracting mixei measures, it may be necessary to convert units.

We

Example

1

Subtract:

5

Solution

1

days 10 hr 15 min day 15 hr 40 min

Before we can subtract minutes, we must convert 1 hour fc 60 minutes. We combine 60 minutes and 15 minutes, makin 75 minutes. Then we can subtract. 9

5 1

—

/-

*

(60 min)

75

9

days yS hr 15 min day 15 hr 40 min

5

-

1

days yS hr y> min day 15 hr 40 min 35

Next

we

convert

1

day

to

min

24 hours and complete th

subtraction. (24 hr)

4

t

9

33 75

4

$ days yS hr >5 min 1 day 15 hr 40 min

35

min

-

1

75

days yS hr y> min day 15 hr 40 min

3 days 18 hr 35

min

Lesson 56

Example

2

Solution

Subtract: 4

We

3 in.

-

2

yd

1 ft 8 in.

numbers with

carefully align the

yard to

1

yd

like units.

We

convert

3 feet.

O

3

^ yd - 2 yd

we

convert

ft)

3 in. 1

8 in.

ft

12 inches. and 3 inches, making 15 inches. Then

Next,

379

1 foot to

combine 12 inches

we

can subtract.

15

2

3

We

I yd t ft - 2 yd 1 ft 1 yd 1ft

t in. 8 in. 7 in.

.ESSON PRACTICE Practice set*

Subtract: a.

-

c.

2

3

hr

1

hr 15

days

3 s

3

min 55

hr 30

-

s

min -

8

b.

1

day

3

8 hr 45

yd yd

1

ft

5 in.

2

ft

7 in.

min

VIIXED PRACTICE

Problem set

1. (31,35)

Three hundred twenty-nine ten-thousandths is how much greater than thirty-two thousandths? Use words to write the answer.

2. {54}

3.

Use

box to solve this problem. The the width of the rectangle is 4 to 3.

a ratio

length to of the rectangle

is

12

what

is its

width?

(b)

what

is its

perimeter?

lot

If

the length

feet,

(a)

The parking

ratio of the

charges $2 for the

first

hour plus 500

for

{m each additional half hour or part thereof. What is the total charge for parking a car in the lot from 11:30 a.m. until 2:15 p.m.? 4. (55)

After four tests Trudy's average score was 85. If her score is 90 on the fifth test, what will be her average for all five tests?

380

Saxon Math 8/7

5.

6. 136.48)

7. (40)

Twelve ounces of Brand X costs $1.50. Sixteen ounces of Brand Y costs $1.92. Find the unit price for each brand. Which brand is the better buy? Five eighths of the rocks in the box were metamorphic. rest were igneous. [a)

What

b)

What was

c)

What percent

fraction of the rocks

were igneous?

the ratio of igneous to of the rocks

metamorphic rocks?

were metamorphic?

Refer to the figure at right to answer a) a)

and

(b).

Name two

of

pairs

vertical

'

angles.

7

b)

Name two

a)

Write six hundred ten thousand in scientific notation.

b)

Write 1.5 x 10 4 in standard form.

8.

angles that are supplemental to

ZRPS.

(51)

9.

Use unit multipliers

to

perform the following conversions:

(50)

a)

216 hours

to

days

minutes

to

seconds

b) 5

10.

a)

Write ^ as a decimal number rounded to the nearest hundredth.

b)

Write | as a percent.

(43, 48)

11. (51)

12.

How many answer in

pennies equal one million dollars? Write the

scientific notation.

Compare: 11 million

O 11

x 10 6

(51)

13. (27)

14. 2,54)

Which even 5

and

two-digit

number

is

a

common

multiple of

7?

There are 100° on the Celsius scale from the freezing temperature to the boiling temperature of water. There 180° on the Fahrenheit scale between these are temperatures. So a change in temperature of 10° on the Celsius scale is equivalent to a change of how many degrees on the Fahrenheit scale?

381

Lesson 56

Refer to the figure below to answer problems 15 and 16. Dimensions are in millimeters. All angles are right angles. 20

30 15 15

10

15.

What

is

the perimeter of the figure?

What

is

the area of the figure?

(19)

16. (37)

Solve:

48 (39)

18.

c

^.O

k -

0.75 = 0.75

(35)

Simplify: 19. 15

2

-

5

3

- VlOO

20. 6 + 12

(20)

21.

m

-r

3

•

2

-

3

(52)

5

+

2

yd yd

2 2

88km .

(53)

1

25.

3-

(26)

4

hr

ft ft

3 in.

22. 1561

9 in.

4hr

24. (23,30)

1 1 2^ - 3-

2

27. Describe

26.

8

(26)

how to

find the 99th

5

-

2

yd yd

2- + 4

2

ft

3 in.

2

ft

9 in.

5f

6

1

111

3- t 24

2

number

38

in this sequence:

(2)

1, 4, 9, 16, 25,

28. (17)

29. iinv. 2, 32)

30. dnv. si

...

Use a protractor and a straightedge to draw a triangle that has a right angle and a 30° angle. Then measure the shortest and longest sides of the triangle to the nearest millimeter. What is the relationship of the two measurements? If

the diameter of a wheel is 0.5 meter, then the radius of wheel is how many centimeters?

A

and B (4, 2). By inspection, find the p 0mt halfway between points A and B. What are the

Graph points

(0, 0)

coordinates of this "halfway" point?

382

Saxon Math 8/7

LESSON

57

i

Negative Exponents • Scientific Notation for Small Numbers

WARM-UP Facts Practice: Powers and Roots (Test K)

Mental Math: a.

40 x 3.2

b.

J 4.2 x 10

c

4 20

d.

Convert 500

*

-

= 2

5

-

3

15

g.

Start

with the number of pounds in a ton,

-

-f

11,

t

f.

-T-

2,

L.

2,

-

I of $25.00

e.

5

mL to -r

1,

v

9,

2.

-f-

Problem Solving: Along the road on which Jesse lives are telephone poles spaced 100 feet apart. If Jesse runs from the first pole to the seventh pole, how many feet does he run? Draw a picture that illustrates the problem.

NEW CONCEPTS Negative

exponents

Cantara multiplied 0.000001 by 0.000001 on her scientific calculator. After she pressed Si the display read

in

-«

X ||J

The

calculator displayed the product in scientific notation. Notice that the exponent is a negative number. So 1

x 10

-12

= 0.000000000001

Studying the pattern below may help us understand the meaning of a negative exponent. i

ltf

10

i

i

yS

ys

10

i

i

i

i

i

10

:

•

10

= 10 3 = 1000

15! :

•

10

= io 2 = 100

ys i

1

in

10

i

10;

10

•

:

i

ill lii

ys

>

ys

ys

•

ys

•

ys

-

ys

10

= 10 1 = 10

Lesson 57

383

Notice that the exponents in the third column can be found

by subtracting the exponents in the first column (the exponent of the dividend minus the exponent of the divisor).

Now we will

continue the pattern. 2

10 3

K)

1U

= 10° •

i

yo

yo

i

i

10

1

i

i

10 2

= 10

3

=

ys

-1

=

10

•

10

10

100

10'

i

i

i

10 1 10

= 10~ 2 =

3

ys

•

10

10

•

i

Notice especially these results: 10° = 1

10"

1

= 10 1

10" z =

10

The pattern suggests two

:

about exponents, which

facts

we

express algebraically below.

If

a

number a

is

not zero, then

a-° = a

n

1

= a'

Example

1

Solution

Simplify: 3" 2

10" 3

(a)

2°

(a)

The exponent

(b)

We rewrite the expression using the reciprocal of the base

(b) is

(c)

zero and the base

with a positive exponent. Then

we

not zero, so 2° equals

1.

simplify.

I 2"9 3

_ - _L

q-2

(c)

is

Again we rewrite the expression with the reciprocal of the base and a positive exponent. 10" 3 =

1

1

10^

1000

(or 0.001)

Saxon Math 8/7

Scientific

notation for small

numbers

the beginning of this lesson, negative exponent can be used to express small numbers in scientific notation 10" 2 meters. If we multiply For instance, an inch is 2.54 x

As we saw

2

54 Dv 10

at

~2

tne P rocmct

'

is

0-0 254

-

2 2.54 x 10" = 2.54 x

10

= 0.0254

2

Notice the product, 0.0254, has the same digits as 2.54 bu with the decimal point shifted two places to the left and witf zeros used for placeholders. The two-place decimal shift tc the left is indicated by the exponent -2. This is similar to the method we have used to change scientific notation tc standard form. Note the sign of the exponent. If the exponenl is a positive number, we shift the decimal point to the right to express the number in standard form. In the number 7 6.32 x 10

the exponent

is

positive seven, so

seven places to the

right.

60200000.

—

we

shift the

decimal point

63,200,000

7 places If

the exponent

point to the

is

a negative number,

write the

left to

number

we

shift the

decimal

in standard form. In the

number 6.32 x 10 the exponent

negative seven, so seven places to the left. is

.000000032

—

-7

we

shift the

decimal point

0.000000632

7 places

we

In either case,

Example 2 Solution

use zeros as placeholders.

Write 4.63 x 10~ 8 in standard notation.

The negative exponent eight places

to

standard form.

the

We

indicates that the decimal point

when

number

written in shift the decimal point and insert zeros as left

the

placeholders.

.00000004,63

—-

0.0000000463

8 places

Example

3

is

Write 0.0000033 in scientific notation.

is

3 85

Lesson 57

Solution

We place the

decimal point

to the right of the first digit that is

not a zero.

0000003.3 6 places

form the decimal point is six places have placed it. So we write

In standard

where we

3.3 x

10

to the left of

-6

Example 4

Compare: zero

O

1

3 x 1°~

Solution

The expression

1

x

10~ 3 equals 0.001. Although this

is less

than

1. it is still

positive, so

it is

number

greater than zero.

zero < 1 x 10" 3

Very small numbers may exceed the display capabilities of a calculator. One millionth of one millionth is more than zero, but it is a very small number. On a calculator we enter

The product, one

more

than can be displayed by many calculators. Instead of displaying one trillionth in standard form, the calculator displays the number in a modified form of scientific notation such as trillionth, contains

or

digits

perhaps I.

x |/J

LESSON PRACTICE Practice set*

Simplify: a.

5

-2

b.

3°

c.

10"

f.

0.000105

i.

5 1.25 x 10~

Write each number in scientific notation: d.

0.00000025

e.

0.000000001

Write each number in standard form: g.

4.5 x 10"

h.

1

x

10-

Compare: j.

i

x

io

_3

1

x

1q:

k.

2.5

x

10" 2

O

2.5

x

10

-3

386

Saxon Math 8/7

MIXED PRACTICE Problem set

1. 1541

.

Use a

ratio

(55)

to

The

solve this problem.

walkers to riders was 5 to

many were 2.

box

j

3. If

ratio

of

315 were walkers, how

riders?

After five tests Allison's average score was 88. After six tests her average score had increased to 90. What was her score on the sixth test?

3. 1281

When

Richard rented a car, he paid $34.95 per day plus 18(2 per mile. If he rented the car for 2 days and drove 300 miles, how much did he pay?

4. If

lemonade

costs $0.52 per quart,

what

is

the cost per pint?

(16)

5. (22)

Diagram

this

statement.

Then answer

the questions

that follow.

Tyrone finished his math homework in two of an hour. (a)

How many

minutes did

it

fifths

take Tyrone to finish his

math homework? (b)

6.

What percent of an hour did his math homework?

it

take Tyrone to finish

Write each number in scientific notation:

(51, 57)

(a)

7.

186,000

(b)

0.00004

Write each number in standard form:

(51, 57)

(a)

8.

1 3.25 x 10

(b)

6 1.5 x 10"

Simplify:

(5?>

(a)

9.

2" 3

(b)

Use a unit multiplier

5°

to convert

(c)

2000

10" 2

I

milliliters to liters.

(50)

10. l21,36)

What

is

the probability of rolling a composite

one toss of a die (dot cube)?

number on

Lesson 57

11. 1461

3 87

Hie tickets for two dozen students to enter the amusement park cost $330. What was the price per ticket? below shows how many students certain intervals on the last test. Create a

12.

The frequency

nv 5)

scored in histogram that illustrates the data in the frequency table.

table

Student Test Scores

% Correct 91-100

W

81-90

jhi

71-80

W

61-70

13.

Frequency

Tally

7

ll

9

mi

6

I

3

in

Compare:

O

(a)

2 2.5 x 10"

(b)

one millionth

(c)

3°

O

2.5

O

-f

10 2 x 10

1

~

2°

Refer to the figure below to answer problems 14 and 15. Dimensions are in yards. All angles are right angles.

1.5

14.

What

is

the perimeter of the figure?

What

is

the area of the figure?

(19)

15. (37)

16. Evaluate:

4ac

if

a = 5 and c = 0.5

(41)

17. Estimate the quotient: $19.89

-f

3.987

(33)

i

18. In the following equation, l41)

3

and

x.

Find y when x

y is

5

more than the product

is 12.

y = 3x +

5

,

of

388

Saxon Math 8/7

Simplify: 19. 20

2

+ 10 3 - V36

21.

3

-

1

yd yd

ft

1 in.

2

ft

3 in.

5^ + 3§ + 6

m

24.

8

2-^-

26.

12

00)

2 27. (4.6 x 10" ) + 0.46 (57)

29. 0.24 x 0.15 x 0.05 (35)

22.

16 oz

(sot

f5«y

2

lpt

23. 48 oz

25.

48 + 12

4-

2

+ 2(3)

(52)

(20)

1561

20.

4 gal 3 qt 1 pt 6 oz

+

1 gal 2 qt 1 pt 5

5|

7t1-

^ 20

36

28. 10

4y

-

(2.3

-

0.575)

(35)

30. 10

4-

(0.14 + 70)

oz

3 89

Lesson 58

E S S

O N

Line Symmetry • Functions, Part 1 WARM-UP Facts Practice: Powers and Roots (Test K)

Mental Math: a.

50 x 4.3

c.

3x 3

5

b. 4.2

= 40

+ 10

d.

2

-r

10 3

Convert 1.5 kg to

e.

10

g.

Find the sum, difference, product, and quotient of

f.

g.

| of $33.00 1.2

and

0.6.

Problem Solving: Four friends met at a party and shook one another's hands. How many handshakes were there in all? Draw a diagram to illustrate the problem. (Students may want to act out the story and count the handshakes.)

NEW CONCEPTS Line

symmetry

A

two-dimensional figure has line symmetry if it can be divided in half so that the halves are mirror images of each other. Line r divides this triangle into two mirror images; so the triangle is symmetrical, and line r is a line of symmetry. r

Actually, the regular triangle has three lines of symmetry.

x

390

Saxon Math 8/7

Example

1

Solution

Draw

a regular quadrilateral

and show

all lines

A

A

regular quadrilateral is a square. square has four lines of symmetry.

of symmetry.

>

symmetry for the figure below. Notice that corresponding points on the two sides of the figure are the same distance from the line of symmetry.

The

y-axis

is

a line of

I

f

(-3, 0)

—

(3, 0)

were folded along the y-axis, each point of the figure on one side of the y-axis would be folded against its corresponding point on the other side of the y-axis. If this

figure

Activity: Line

Symmetry

Materials needed: • 1.

Paper and scissors

Fold a piece of paper in

folded

Beginning and ending

half.

edge

at

the folded edge, cut a pattern out

of the folded paper.

Open

the cut-out and note

its

symmetry.

Lesson 58

391

Fold a piece of paper twice as shown.

2.

four corners

two corners

Hold the paper on the corner opposite the "four corners," and cut out a pattern that removes the four corners. hold here

sample cut pattern

four corners

How many

Unfold the cut-out.

lines of

symmetry do

you see?

Functions, part 1

A

function

certain rule.

is

a set of

We

number

pairs that are related

will study pairs of

numbers

to

by

a

determine a

and then we will use the rule to find the missing number. Note that for a function there is exactly one "out" number for every "in" number.

rule for the function,

Example 2

Find the missing number.

jN

q ut

Ppl

3

U N

5

C

15

° N

30

*~

— — 10—— Solution

We

9

study each "in-out" number pair to determine the rule for the function. We see that for each complete pair, if the "in" number is multiplied by 3, it equals the "out" number. Thus the rule of the function is "multiply by 3." We use this rule to find the missing number. We multiply 7 by 3 and find that the missing

number

is

21.

392

Saxon Math 8/7

LESSON PRACTICE Practice set

a.

b.

Copy this and show The

rectangle on your paper, lines of

its

y-axis

symmetry.

a line of

is

symmetry

for a triangle.

coordinates of two of its vertices are (0, 1) and What are the coordinates of the third vertex?

The

(3, 4).

Find the missing number in each diagrajn: c.

iN

4-

U N

3-

C

T 7-

I

O N

9-

d. i N

Out

F

— — — —

F

Out 4

20

0-

U N

15

1-

C

T 35-

45

7

I

O N

9

MIXED PRACTICE Problem set

1. (28, 35)

from Jim's house to school. How far does Jim walk going to and from school once every day 1.4 kilometers

It is

for 5

2. (28)

The parking

each half hour or part of a half hour. If Edie parks her car in the lot from 10:45 a.m. until 1:05 p.m., how much money will she pay?

3. If (41)

4.

days?

lot

charges

the product of the

sum

of the

The

football

75(2 for

number n and 17

is

340,

what

is

the

number n and 17? team

won

3 of its 12

games but

lost the rest.

(36)

(a)

What was

(b)

What

(c)

What percent

fraction of the

of the

games did the team

lose?

games did the team win?

bowling average after 5 games was 120. In his next 3 games, Willis scored 118, 124, and 142. What was Willis's bowling average after 8 games?

5. Willis's (55)

the team's win-loss ratio?

n 3 93

Lesson 58

6. (22)

Diagram

statement.

this

Then answer

the questions

that follow.

Three

fifths of the

60 questions were multiple-

choice. (a)

How many

(b)

What percent

of the 60 questions

were multiple-choice?

of the 60 questions

were not multiple-

choice? 7. (Inv. 2)

In the figure at right, the center of

the circle

is

point

O and OB

= CB.

Refer to the figure to answer (a)-(d). (a)

Name

(b)

Name two

three radii.

chords that are not

diameters.

8.

(c)

Estimate the measure of central angle BOC.

(d)

Estimate the measure of inscribed angle BAC.

Write each number in standard form:

151,571

9.

(a)

1.5

(c)

10" 1

x 10 7

Compare: 20

(16)

(b)

qt

O

2.5 x 10

-4

5 gal

by 0.18 and write the answer rounded neareS { w hole number.

10. Divide 3.45 03, 45)

11.

Find the next three numbers in

to the

this sequence:

(2)

20, 15, 10, 12. (48)

Complete the

table.

i

6

(c)

13. (58)

Find the missing number.

Percent

Decimal

Fraction

(b)

(a)

16%

(d)

F

In

6

— — — — —

2

O

Out

0^

U N

3

C T

12

—N—

24

I

394

Saxon Math 8/7

measure of ZD is 35° and the measure of ZCAB is 35°. Find the measure of

14. In the figure at right, the

m

15. (inv.3,58)

16. 1581

(a)

ZACB.

(b)

ZACD.

(c)

ACAD.

The

y-axis

is

a line of

coordinates of two of

its

symmetry

for a triangle.

and

vertices are (-3, 2)

(0, 5).

(a)

What

are the coordinates of the third vertex?

(b)

What

is

A

the area of the triangle?

regular pentagon has

lines of

how many

symmetry?

how

17. (a) Traveling at 60 miles per hour, (53)

The

long would

take to travel 210 miles?

How

(b)

would

long

the

same

trip take at

70 miles per

hour? Solve: 18. (39)

15 -W ^ =

7 5

- y =

19. 1.7

Z

0.17

(35)

Simplify: 3

20. 10

- 10 2 + 10 1 - 10°

(20, 57)

24

.

(50)

1 gal 2 qt 1 pt

+ lgal 2 qt

1 pt

2mi-^A mi ( V

a 2- + 5a

5

23. (56)

25.

1

26. (26, 30)

- 4 -

21. 6 + 3(2)

(5

+

(5 2)

22. (49)

it

day

1

-

8

10 -

(£4 V

(23, 30)

\

2j

-r

2-

27.

3-

5

(26)

4

•

(

min hr 30 min

3 hr 15

- 1* 6

i\

3)

Lesson 58

28. Evaluate: b

2

- Aac

if

a = 3.6, b =

6,

and

395

c = 2.5

(52)

29. (4,

(a)

Arrange these numbers in order from greatest to

least:

10)

^' 2'

(b)

30.

Which

of the

numbers

in

2"'

^

are integers?

(a)

Lindsey had the following division

to perform:

(27)

1

35

22

how

Lindsey could form an equivalent division problem that would be easier to perform mentally.

Describe

396

Saxon Math 8/7

LESSON

59

Adding Integers on the

Number Line

WARM-UP Facts Practice: Fraction-Decimal-Percent Equivalents (Test L)

Mental Math: a.

60 x 5.4

c.

§§

e.

g.

2

= •

2

f 3

b.

2 0.005 x 10

d.

Convert 185

m.

| of $40.00

f.

At $7.50 an hour,

cm to

how much money can

Shelly earn in 8 hours?

Problem Solving:

A

square and a regular pentagon share a common side. If the perimeter of the pentagon is one meter, what is the area of the square in square centimeters?

NEW CONCEPT whole numbers and

Recall that integers include all the

also

the opposites of the positive integers (their negatives). All the

numbers

in this sequence are integers: .

The dots on

.

this

.

,

3

,

2

number

1

,

0, 1

,

line

,

2

,

mark

3,

...

the integers from -5

through +5:

-5

Remember

-4

-3

that the

such as 3| and

-2

-1

1

2

3

4

5

numbers between the whole numbers,

1.3, are

not integers.

numbers on the number line except zero are signed numbers, either positive or negative. Zero is neither positive nor negative. Positive and negative numbers have a sign and a value, which is called absolute value. The absolute value of a number is its distance from zero. All

Numeral

Number

+3

Positive three

-3

Negative three

Sign

Absolute Value

+

3 3

1

Lesson 59

397

The absolute value of both +3 and -3 is 3. Notice on the number line that +3 and -3 are both 3 units from zero. We may use two vertical segments to indicate absolute value. = 3

|3|

Example

1

Solution

=

|-3|

3

The absolute value

The absolute value

of 3 equals

of -3 equals

Simplify:

1

3

—

3.

3.

5

To find the absolute value of 3 - 5, we first subtract 5 from 3 and get -2. Then we find the absolute value of -2, which is 2. Absolute value can be represented by distance, whereas the sign can be represented by direction. Thus signed numbers are sometimes called directed numbers because the sign of the number (+ or -) can be thought of as a direction indicator.

When we add, numbers, we need

multiply,

divide directed to pay attention to the signs as well as the absolute values of the numbers. In this lesson we will subtract,

practice adding positive

or

and negative numbers.

A number line can be used to illustrate the addition of signed numbers. A positive 3 is indicated by a 3-unit arrow that points to the right. A negative 3 is indicated by a 3-unit arrow that points to the

left.

-H

I

h-

1

-3

+3

To show the addition of +3 and -3, we begin at zero on the number line and draw the +3 arrow. From its arrowhead we draw the -3 arrow. The sum of +3 and -3 is found at the point on the number line that corresponds to the second arrowhead. -3

i

i i

+3

——

I

-5

1

1

1

-4

-3

-2

We see that the sum of +3 opposites

is

always zero.

1

-1

1

@

and -3

1

1

is 0.

1

1

1

2

3

4

1

5

We find the sum of two

Saxon Math 8/7

Example

2

(b)

+ (+5)

(-3)

(a)

number

addition problem on a

Show each

line:

(-4) + (-2) j

Solution

(a)

We

begin

at

draw an arrow 3 units long the From this arrowhead we draw an arro^

zero and

points to the left. 5 units long that points to the right. of -3

and +5

We

see that the sui

is 2.

+5

-3

-4

-5

(b)

-1

-2

-3

We use arrows to show that the sum of -4

and -2

is

-6.

-2

-4 -6

Example

3

Show

-4

-5

this addition

problem on a number (-2)

Solution

This time

we draw

-1

-2

-3

line:

+ (+5) + (-4)

!

We

always begin the first begin each remaining arrow at the

three arrows.

arrow at zero. We arrowhead of the previous arrow.

-4 +5 -2

— -5

The

-4

-3

i

i

i

i

u

-2

arrowhead corresponds to -1 on the number the sum of -2 and +5 and -4 is -1.

Example 4

last

line, so

The troop began the hike on the desert floor, 126 feet below sea level. The troop camped for the night on a ridge 2350 feet above sea level. What was the elevation gain from the start of the hike to the campsite?

Lesson 59

Solution

A number

399

oriented vertically rather than horizontally is more helpful for this problem. The troop climbed 126 feet to reach sea level (zero elevation) and then climbed 2350 feet more to the campsite. We calculate the total elevation gain as shown below. line

that

is

- +2350

ft

2000

1000

sea

Example

5

- -126

feet

2476

feet

ft

feet

any money. In order to buy a friend's birthday present, Krissie borrowed $5 from her sister. Later Krissie received a check for $25 from her grandmother. After Krissie did not have

she repays her Solution

level

126 + 2350

We may

sister,

how much money will

Krissie have?

use negative numbers to represent debt (borrowed

money). After borrowing $5, Krissie had negative five dollars. Then she received $25. We show the addition of these dollar amounts on the number line below.

+25 -5

-20

-15

-10

-5

!

10

30

25

15

After she repays her sister, Krissie will have $20.

LESSON PRACTICE Practice set

Use arrows

to

show each addition problem on

a

a.

(-2)

+ (-3)

b. (+4)

+ (+2)

c.

(-5)

+ (+2)

d. (+5)

+ (-2)

e.

(-4)

+ (+4)

f.

(-3)

number

+ (+6) + (-1)

line:

400

Saxon Math 8/7

Simplify: g.

j.

|-3

+

1

h.

|3|

-

|3

i.

3|

|5

-

3|

}

j

the return trip the troop hiked down the mountai from 4362 ft above sea level to the valley floor 126 below sea level. What was the drop in elevation durin

On

1

the return trip?

k.

did not have any money. In order to buy a movii ticket, he borrowed $5 from his brother. Sam wants t( earn enough money to repay his brother and to buy a $21 ticket to the amusement park. How much money doe.' Sam need to earn?

Sam

MIXED PRACTICE Problem set

1. [28]

School pictures cost $4.25 for an 8-by-10 print. They cost $2.35 for a 5-by-7 print and 60(2 for each wallet-size print.

What

is

the total cost of

two 5-by-7 prints and

six wallet-

size prints?

The double-line graph below compares the daily maximum temperatures for the first seven days of August to the average

maximum

temperature for the entire month of August. Refer to the graph to answer problems 2 and 3: Maximum Temperature Readings August 1-7, 1998

for Tri-City Area,

80

1

'

2

1

2. (38 '

3. '

1

4 Date

I

1

1

5

6

7

The maximum temperature reading on August

6,

how much

temperature

for the

(28 38)

1

3

greater than the average

month

What was

maximum

1998, was

of August?

the average

maximum

seven days of August 1998?

temperature for the

first

Lesson 59

4. 591

5. (54>

On January

noon was 7°F. By 10 p.m. the temperature had fallen to -9°F. The temperature dropped how many degrees from noon to 10 p.m.? Use

1

the temperature at

problem. The ratio of sonorous to discordant voices in the crowd was 7 to 4. If 56 voices were discordant, how many voices were sonorous? a ratio

box

6.

Diagram

f

that, follow.

The

to solve this

statement.

this

Celts

won

Then answer

the questions

three fourths of their first 20 games.

(a)

How many of their first

(b)

What percent to

7.

401

20 games did the Celts win?

of their first 20

games did the

Celts fail

win?

Compare:

O

|-3|

|3|

(59)

8.

(a)

Write 4,000,000,000,000 in scientific notation.

(b)

Pluto's average distance from the

(51)

Sun

is

9 3.67 x 10

miles. Write that distance in standard form.

9.

(a)

A

micron

is

x 10

1

6

meter. Write that

number and

unit in standard form. (b)

10.

Compare:

1

millimeter

Use a unit multiplier

O

1

to convert

3 x 10~

300

meter

mm to m.

(50)

11.

Complete the

table.

(48)

Fraction

1

Use arrows

number (a)

(+2)

to

12% (d)

(c)

3

(59)

show each addition problem en

line:

+

(-5)

Percent

(b)

(a)

12.

Decimal

(b)

(-2)

+ (+5)

a

—

-

402

i

Saxon Math 8/7

13.

Find the missing number.

Out

F

In

—

(58)

2

12 8

u N

— —

C

T

—

14

—-

24

— — LJ 1

— ON — I

-

12

Refer to the figure below to answer problems 14 and 15 Dimensions are in millimeters. All angles are right angles. 50 20 30

60

15

14.

What

is

the perimeter of the figure?

15.

What

is

the area of the figure?

(37)

Solve: 16. 4.4 =

0.8

8w

17.

(35)

18.

n + 11 = VL 20

oo)

19

30

1.5

J-

(39)

0364 _

m

(35)

7

Simplify: 20. 2

_1

+ 2~ x

21.

(57)

V64 -

2

3

+ 4°

(20, 57) \

22.

3

yd

2

ft

7\Z

in.

(49)

23. (56)

+

1

yd

24. 2 1 hr 2 r5j;

•

5|

^Omi 1

hr

in.

25. r^;

1 qt 1 pt

6 oz

1 pt

12 oz

-

f

1

| 9

•

12] )

-f

63

403

Lesson 59

Estimate each answer to the nearest whole number. Then

perform the calculation. 26. (23, 30)

3^

4-1-

6

27.

9

28. Evaluate: a

(26, 30)

- be

if

a = 0.1. b = 0.2, and c = 0.3

(52)

29. (46)

Find the tax on an $18.00 purchase when the sales-tax rate is 6.5%.

30. This table (36)

student

shows the

who

voted

results of a class election. If

is

selected at random,

what

is

probability that the student voted for the candidate

received the most votes? Vote Tally

Candidate

Vasquez

Lam Enzinwa

Votes

m W IHt

ii

Hit III

one the

who

404

Saxon Math 8/7

LESSON

HntH

Fractional Part of a Number, Part 1 • Percent of a Number, Part 1

WARM-UP M)

Facts Practice: Metric Conversions (Test

Mental Math: 2

a.

70 x 2.3

b.

435

c.

5x -

d.

Convert 75

e.

g.

= 49

1

-r

10

mm to cm.

of $1.00 f. Vl44 - a/25 f Start with 25(2, double that amount, double that amount, double that amount, x 5, add $20, ^ 10, + 10.

Problem Solving:

Copy

this

problem and

fill

in the missing digits: x

_

1101

NEW CONCEPTS We

part of a

problems by translating the question into an equation and then solving the

number,

equation.

Fractional

P ar *

Example

can

solve

To

translate,

we

replace the

word

we

replace the

word o/with

^

1

Solution

fractional-part-of-a-number

What number

is

is

with = x

0.6 of 31?

This problem uses a decimal number to ask the question. We represent what number with N We replace is with an equal sign. We replace o/with a multiplication symbol.

W

What number 1

find the answer,

0.6 of 31?

I

WN To

is

.

1

1

I

= 0.6 x 31

we

WN

question

equation

multiply.

= 18.6

multiplied

Lesson 60

Example 2 Solution

Three

fifths of

120

405

what number?

is

This time the question is phrased by using a common fraction. The procedure is the same: we translate directly.

Three

fifths of

120

is

what number?

III

I

-

I

WN

x 120 =

3

To

we

find the answer,

question

equation

multiply.

WN

= 72

We

part

1

can translate percent problems into equations the same way we translate fractional-part-of-a-number problems: we convert the percent to either a fraction or a decimal.

Example

3

The

Percent of a

number,

jacket sold for $75. Forty percent of the selling price

profit.

Solution

We

How much money is 40%

of $75?

an equation.

translate the question into

the percent to a fraction or to a decimal.

Example 4

A

40 100

=

WN

2 = — X $75

WN

= $30

x

$75

5

how much

is

We want to find 8%

convert both ways.

WN

= 0.40 x $75

WN

= 0.4 x $75

WN

= $30

commission of

8%

the salesperson sells a car for the salesperson's commission?

of the selling price of a car.

Solution

We show

certain used-car salesperson receives a

$3600,

We may

Percent to Decimal

Percent to Fraction

WN

was

If

of $3600. This time

we convert the percent

to a decimal.

Eight percent of $3600 I

I

0.08

x

I

is

commission.

I

I

C

$3600 = $288 =

The salesperson's commission

is

C

$288.

406

Saxon Math 8/7

Example

5

Solution

What number This time

we

is

25%

of 88?

convert the percent to a fraction.

What number

is

25%

(

i

i

WN

=

\

WN Whether

a percent should be

of 88? I

I

x 88

I

= 22

changed

to a fraction or to a

decimal is up to the person solving the problem. Often one form makes the problem easier to solve than the other form. With practice the choice of which form to use becomes more apparent.

LESSON PRACTICE Practice set*

Write equations to solve each problem: a.

What number

b.

Three eighths of 3|

c.

What number

d.

Seventy-five hundredths of 14.4

e.

What number

f.

g.

is

is

is

of 71? | b is

6 of 145?

50%

Three percent of $39

What number

is

what number?

25%

is

what number?

of 150?

is

how much money?

of 64?

commission of 12% of sales, the salesperson's commission on $250,000 of sales?

h. If a salesperson receives a

what

is

MIXED PRACTICE Problem set

1. (31, 351

2.

m

Five and seven hundred eighty-four thousandths is how much less than seven and twenty-one ten-thousandths?

Cynthia was paid 200 per board for painting the fence. If she was paid $10 for painting half the boards, how many boards were there in all?

407

Lesson 60

3.

-

1411

4.

When

divided by n, the quotient product when 72 is multiplied by n?

Four

72

is

fifths of

the students passed the

is 12.

What

is

the

test.

(36)

(a)

What percent

(b)

What was

who 5. (55}

6.

7.

who

passed to students

did not pass?

Write each number in scientific notation: , ,

0.00000008

(a)

-

the ratio of students

The average height of the five players on the basketball team was 77 inches. One of the players was 71 inches tall. Another was 74 inches tall, and two were each 78 inches tall. How tall was the tallest player on the team?

(51, 57)

(22 48)

of the students did not pass the test?

Diagram

(b)

this statement.

67.5 billion

Then answer

the questions that

follow.

Two

thirds of the

(a)

How many

(b)

What percent

96 members approved of the plan.

of the 96

of the

members approved

of the plan?

members did not approve

of the

plan?

8. l59)

The

first

stage of the rocket fell from a height of 23,000

feet

and

settled

level. In all,

on the ocean

how many

feet

floor

9000

feet

did the rocket's

below sea first

stage

descend? Write equations to solve problems 9 and 10. 9.

What number

is

f of 17?

(60)

40% of the selling price of a $65 how many dollars profit does the

10. If

m

sweater

11.

is

sweater store

is profit,

make when

sold?

Compare:

(43, 59)

(a)

^00.33 o

(b)

|5

-

3|

then

O

|3

-

5|

the

408

Saxon Math 8/7

12.

Complete the

table.

Fraction

Decimal

Percent

(48) i

8

13. l59)

Use arrows

number (a)

(-3)

to

125%

(d)

(c)

show each addition problem on

a

line:

+ (-1)

14. (a) Write the

(b)

(-3)

+ (+1)

prime factorization of 3600 using exponents.

(21)

(b)

15. (Inv 51 -

Write the prime factorization of V3600.

Find the number of degrees in the central angle of each sector of the circle at right.

Refer

to

the

figure

below

to

answer problems

Dimensions between labeled points are in of ZEDF equals the measure of ZECA.

6

16. (a)

F

feet.

16-18.

The measure

6

Name

a triangle congruent to triangle DEF.

Name

a triangle similar to

(is)

(b)

ADEF but

not congruent

to

A DEF. 17. (a)

Find the area of ABCD.

(37)

(b)

18. l37)

Find the area of AACE.

By

subtracting the areas of the two smaller triangles from the area of the large triangle, find the area of the quadrilateral

ABDF.

Lesson 60

409

Solve: 19.

20.

p

30

(30)

20

9m

= 0.117

(35)

Simplify: 21. 3

2

+ 4(3 +

2)

-

2

3 •

2" 2 + \36

52;

22.

m

days 16 hr 48 min day 15 hr 54 min

3

+

1

3

24. 3^ 5 (30) 26. 6.5

- (5

^

6

)

-

(0.65

23.

m

25.

m

30. dnv.

3.

ss)

14

- 0.065)

6

8

-r

— 12

27. 0.3 ^ (3

+ 24

^ 0.03)

(45)

Use (1

m

a unit multiplier to convert 3.5 centimeters to meters.

= 100 cm)

why

these two division problems are equivalent. Tn en give a money example of the two divisions.

29. Explain (27,45)

4

(26)

(35)

28.

7

19^ + 27 - + 24^

The

x-axis

is

a

1.5

150

0.25

25

line

of

symmetry for AABC. The (3, 0), and the coordinates of

coordinates of point A are point B are (0, -2). Find the coordinates of point C.

410

Saxon Math 8/7

Focus on Classifying Quadrilaterals 18 that a four-sided polygon is quadrilateral. Refer to the quadrilaterals shown below answer the problems that follow.

Recall

from Lesson

a to

D

1.

Which

figures

have four right angles?

2.

Which

figures

have four sides of equal length?

3.

Which

figures

have two pairs of parallel sides?

4.

Which

figure has just

5.

Which

figures

have no pairs of parallel sides?

6.

Which

figures

have two pairs of equal-length sides?

We

can

sort quadrilaterals

one pair of parallel sides?

by

their characteristics.

by the number of pairs of with two pairs of parallel sides sort is

show

7.

parallel sides. is

One way

to

A quadrilateral

a parallelogram. Here

we

four parallelograms.

Which

of the figures

A

A-G are

parallelograms?

quadrilateral with just one pair of parallel sides is a trapezoid. The figures shown below are trapezoids. Can you find the parallel sides? (Notice that the parallel sides are not

the

same

length.)

1

41

Investigation 6

Which

8.

A

of the figures

quadrilateral with

Here

Which

9.

We

we show two

no

A-G is

a trapezoid?

pairs of parallel sides

is

a trapezium.

examples:

of the figures

A-G are

trapeziums?

can sort quadrilaterals by the lengths of their sides.

four

sides

the

are

equilateral.

An

rhombus

a

is

same

If

the

the quadrilaterals are equilateral quadrilateral is a rhombus. A length,

type of parallelogram. Here

we show two

examples.

10.

Which

of the figures

A-G are rhombuses?

We

can sort quadrilaterals by the measures of their angles. If the four angles are of equal measure, then each angle is a right angle,

and the quadrilateral

is

a rectangle.

A

rectangle

is

a

type of parallelogram. 11.

Which

of the figures

A-G are

rectangles?

Notice that a square is both a rectangle and a rhombus. A square is also a parallelogram. We can use a Venn diagram to illustrate the relationships.

parallelogram

Saxon Math 8/7

Any

within the circle labeled "rectangle" is a parallelogram as well. Any figure within the circle labeled "rhombus" is also a parallelogram. A figure within both the figure that

rectangle

12.

Copy

is

and rhombus

the

circles is a square.

Venn diagram above on your

Draw

Then

refer

E at

the beginning of this each of the quadrilaterals in the Venn

to quadrilaterals A, B, C, D,

investigation.

and

paper.

diagram in the proper location. (One of the figures will be outside the parallelogram category.)

A

student made a model of a rectangle out of straws and pipe cleaners (Figure J). Then the student shifted the sides so that two angles became obtuse and two angles became acute (Figure K).

\///////////////////////////A

W/////////////////////////A

W/////////////////////////A

\///////////////////////////A

Figure J

Refer to Figures

J

and

Figure

K to answer problems

13. Is Figure

K

a rectangle?

14. Is Figure

K

a parallelogram?

15.

Does the perimeter of Figure Figure

16.

K

K

13-16.

equal the perimeter

of

J?

Does the area of Figure

K

equal the area of Figure

J?

Another student made a model of a rectangle out of straws and pipe cleaners (Figure L). Then the student reversed the positions of two of the straws so that the straws that were the

413

Investigations

same length were adjacent

to

each other instead of opposite

each other (Figure M). V//////////////////////////A

V///////J//////////////////A

Figure

M

M

Figure does not have a pair of parallel sides, so it is a trapezium. However, it is a special type of trapezium called a kite. 17.

Which

18. If

of the figures

two sides of

A-G is

a kite?

a kite are 2

ft

and

3

ft,

what

is

the

perimeter of the kite? Notice that a kite has a line of symmetry. 19.

Draw

a kite

20.

Draw

a

and show

rhombus

its

line of

symmetry.

that is not a square,

and show

its

lines of

show

its

lines of

its

lines of

symmetry. 21.

Draw

a rectangle that

is

not a square, and

symmetry. 22.

Draw

a

rhombus

that is a rectangle,

and show

symmetry. 23.

An

isosceles trapezoid has a line of symmetry. nonparallel sides of an isosceles trapezoid are the length.

Draw an

isosceles trapezoid

and show

its

The same

line of

symmetry. every trapezoid has a line of symmetry. Any parallelogram that is not a rhombus or rectangle does not have line symmetry. However, every parallelogram does have

Not

414

Saxon Math 8/7

point symmetry. A figure is symmetrical about a point every line drawn through the point intersects the figure points that are equal distances from the point of symmetry.

if

at

We

can locate the point of symmetry of a parallelogram by finding the point where the diagonals of the parallelogram intersect. A diagonal of a polygon is a segment between nonconsecutive vertices.

point of

In the following

symmetry

problem we learn a way

to test for point

symmetry. 24.

Draw two

or three parallelograms

on grid paper. Be sure rectangle and one is not

one of the parallelograms is a a rectangle. Locate and mark the point in each parallelogram that

where the diagonals parallelograms.

If

we

intersect.

will appear to be in the rotated.

On

Then

with point symmetry point of symmetry, the figure

same position

it

was

in before

it

one of the cut-out parallelograms, place

the tip of a pencil intersect.

carefully cut out the

rotate a figure

a half turn (180°) about its

was

Then

on the point where the diagonals

rotate the parallelogram 180°. Is the point

of intersection a point of

symmetry?

Repeat the rotation with the other parallelogram(s) you cut out.

Investigation 6

25.

Which

Below we

of the figures

A-G have

415

point symmetry?

classify the figures illustrated at the beginning of

this investigation.

You may

refer to

them

as

you answer the

remaining problems.

parallelogram

parallelogram

parallelogram

parallelogram

rectangle

rhombus

rectangle

rhombus square

trapezium

trapezoid

trapezium kite

Answer 26.

A

true or false,

square

is

and

state the reason(s) for

your answer.

a rectangle.

27. All rectangles are parallelograms. 28.

Some

squares are trapezoids.

29.

Some

parallelograms are rectangles.

30.

Draw

a

Venn diagram

illustrating the relationship

quadrilaterals, parallelograms,

and trapezoids.

of

416

Saxon Math 8/7

LESSON

61

Area of a Parallelogram

•

Angles of a Parallelogram WARM-UP Facts Practice: Fraction-Decimal-Percent Equivalents (Test L)

Mental Math: a.

50 x 4.6

c.

;t!t

e.

3

g.

What

ZU

2

—

^tt

-

2

o

3

is

_1

b.

2.4 x 10

d-

Convert 1.5

f.

km to

m.

^ of $3.00

the total cost of a $20 item plus

6%

sales tax?

Problem Solving:

The squares

of the

100. Altogether,

first

nine counting numbers are each less than counting numbers have squares that

how many

are less than 1000?

NEW CONCEPTS Area of a parallelogram

Recall from Investigation 6 that a parallelogram in

which both

is

a quadrilateral

pairs of opposite sides are parallel.

Parallelogram

Parallelogram

Not a parallelogram

lesson parallelograms. In

we

this

scissors to

will

We

finding the areas of paper parallelogram and

practice

can use a help us understand the concept. Activity

Materials needed: •

Paper

•

Scissors

•

Straightedge

1:

iArea of a Parallelogram

7 Lesson 61

Cut a piece of paper may use graph paper

to if

41

form a parallelogram as shown. You available.

Next, sketch a segment perpendicular to two of the parallel sides of the parallelogram. Cut the parallelogram into two pieces along the segment you drew.

two pieces and The area of the

Finally, reverse the positions of the

together

to

form a rectangle.

fit

them

original

parallelogram equals the area of this rectangle.

The dimensions

of a rectangle are often called the length and the width. When describing a parallelogram, we do not use these terms. Instead we use the terms base and height.

Notice that the height is not one of the sides of the parallelogram (unless the parallelogram is a rectangle). Instead, the height is perpendicular to the base. Multiplying the base by the height gives us the area of a rectangle.

418

Saxon Math 8/7

we saw

the area of the rectangle equals the area of the parallelogram we are considering. Thru we find the area of a parallelogram by multiplying its base b}

However,

its

as

in Activity

1,

height.

Area of a parallelogram = base

Example

1

Find the

(a)

perimeter and

(b)

•

height

area of this

parallelogram. Dimensions are in inches. 8

Solution

(a)

We

find the perimeter

The opposite

5 in.

is

+ 8

in.

+ 5

the area

is

8

in.,

= 26

in.

in.

and the height

is

4

in.

So

is

(8 in.)(4 in.)

parallelogram

+ 8

in.

We find the area of a parallelogram by multiplying the base by the height. The base

Angles of a

sides.

sides of a parallelogram are equal in length.

So the perimeter

(b)

by adding the lengths of the

= 32

in.

2

Figures J and K of Investigation 6 illustrated a "straw" rectangle shifted to form a parallelogram that was not a rectangle. Two of the angles became obtuse angles, and the other two angles became acute angles. U//////////////////////////A \///////////////////////////A

W///////////////77777-/77777A

W/////////////////////////A

Figure J

Figure

K

two of the angles became more than 90°, and two of the angles became less than 90°. Each angle became greater than or less than 90° by the same amount. If, by In other words,

shifting the sides of the straw rectangle, the obtuse angles

became 10° greater than 90° (100° angles), then the acute angles became 10° less than 90° (80° angles). The following activity illustrates this relationship.

9 Lesson 61

Activity 2:

41

Angles of a Parallelogram

Materials needed: •

Protractor

•

Paper

•

Two

pairs of plastic straws (The straws within a pair

must be the same

length.

The two

pairs

may

be

different lengths.) •

Thread or lightweight

•

Paper clip

Make

string

for threading the straws (optional)

a "straw" parallelogram

by running

through two pairs of plastic straws.

If

a string or thread

the pairs of straws are

of different lengths, alternate the lengths as

you thread them

(long-short-long-short).

^^////////////////////^^^ Bring the two ends of the string together, pull until the string is snug but not bowing the straws, and tie a knot.

//////////////////////

I ////////////////////// i

You should be

able to shift the sides of the parallelogram to

various positions.

//////////////////777,

V///////////////77777,

7ZZZZZZZZZZZZZZZZMZA

Tzzzmzzzzzzzzzzzmx

Lay the straw parallelogram on a desktop with a piece of paper under it. On the paper you will trace the parallelogram. Shift the parallelogram into a position you want to measure, hold the straws and paper still (this may require more than two hands), and carefully trace with a pencil around the inside of the parallelogram.

420

Saxon Math 8/7

Set the straw parallelogram aside, and use a protractor tc measure each angle of the traced parallelogram. Write the

measure inside each angle. Some groups may wish to trace and measure the angles of a second parallelogram with a different shape before answering the questions below. 1.

What were

the measures of the

two obtuse angles of one

parallelogram? 2.

What were

the measures of the

two acute angles of

the

same parallelogram? 3.

What was

the

sum

of the measures of one obtuse angle

and one acute angle of the same parallelogram? If

you traced two parallelograms, answer the three questions

again for the second parallelogram.

Record several groups' answers on the board. Can any general conclusions be formed?

The

quality of

all

types of measurement

is

affected

by

the

measuring instrument, the material being measured, and the person performing the measurement. However, even rough measurements can suggest underlying relationships. The rough measurements performed in Activity 2 should suggest the following relationships between the quality

of the

angles of a parallelogram: 1.

2.

Nonadjacent angles (angles in opposite corners) have equal measures. Adjacent angles (angles that share a common side) are supplementary that is, their sum is 180°.

—

Example 2

mZD

is

110°.

Find the measures of angles A,

B,

and C

In parallelogram

ABCD,

d /iio° /

in the parallelogram.

c

Solution

a

/ / b

Nonadjacent angles, like ZB and ZD, have equal measures, so mZB = 110°. Adjacent angles are supplementary, and both ZA and ZC are adjacent to ZD, so mZA = 70° and mZC = 70°.

J

Lesson 61

421

LESSON PRACTICE Practice set

Find the perimeter and area Dimensions are in centimeters.

/~

a.

~~7!

b.

each

of

N

n

\

\

!

parallelogram.

\13

12

12 10

c.

10

10

For problems d-g, find the measures of the angles marked e, f,

and g

d,

in this parallelogram.

Figure ABCD is a parallelogram. Refer to this figure to find the measures of the angles in problems h-j.

h.

ZA

i.

ZADB

j.

ZABC

MIXED PRACTICE Problem set

1.

If

\ gallon of

milk costs $1.12, what

is

the cost per pint?

(16, 46)

2. (54>

3. (55)

problem. The cookie recipe called for oatmeal and brown sugar in the ratio of 2 to 1. If 3 cups of oatmeal were called for, how many cups of brown sugar were needed?

Use

a ratio

box

to solve this

Ricardo ran the 400-meter race 3 times. His fastest time was 54.3 seconds. His slowest time was 56.1 seconds. If his average time was 55.0 seconds, what was his time for the third race?

422

Saxon Math 8/7

4.

5. (51)

Paula runs to end of the trail and back in 60 minutes, what is average speed in miles per hour? It is

4| miles to the end of the

6.

the

hei

Sixty-three million, one hundred thousand is how much greater than seven million, sixty thousand? Write the

answer in

(36,48)

trail. If

scientific notation.

Only three tenths of the print area of the newspaper carried news. The rest of the area was filled with advertisements.

What percent

(a)

of the

was

print

area

news

area to advertisement

filled

with

advertisements?

What was

(b)

the ratio of

area? (c)

If,

without looking, Kali opens the newspaper and

places a finger on the page, that her finger will be

7.

what

is

the probability

on an advertisement?

(a)

Write 0.00105 in scientific notation.

(b)

Write 3.02 x 10 5 in standard form.

(51, 57)

8.

Use prime factorization

to

Use a unit multiplier

convert 1760 yards to

reduce

j§|.

(24)

9.

to

feet.

(50)

In

and

t

he figure below,

EC

II

quadrilateral

ABDE

is

a

rectangle

FB. Refer to the figure to answer problems 10-12.

FA

E

D

B

C

10. Classify each of the following quadrilaterals: A

°

(Inv.6)

(a)

ECBF

(b)

ECBA

61

423

AE

= 8 m.

Lesson

11. l37- 61]

If

AB

=

ED

= 4 m.

BC

=

EF

= 6 m. and

BD

-

then (a)

what

is

the area of quadrilateral

(b)

what

is

the area of triangle

(c)

what

is

the area of quadrilateral

12. Classify

BCEF?

ABF? ECBA?

each of the following angles as acute,

right, or

obtuse: (a)

13. llnv"*

ZECB

(b)

ZEDC

(c)

ZFBA

Following is an ordered list of the number of correct answers on a ten-question quiz taken by 19 students. Find the (a) median, (b) first quartile. and (c) third quartile of these scores: (d) identify any outliers. 2. 5. 5. 6. 6. 6. 7. 7. 7. 8. 8. 8. 8. 9. 9. 10. 10. 10.

14. Refer to the parallelogram ffinr. 6.

to

answer

(a)-{c).

61)

16

15. '

below

10

61}

cm

(a)

What

is

the perimeter of this parallelogram?

(b)

What

is

the area of this parallelogram?

(c)

Trace the parallelogram on your paper, and locate point of symmetry.

its

The

parallelogram at right is divided bv a diagonal into two Find the triangles. congruent

measure of (a)

16.

Za.

(b)

Zb.

(c)

Zc.

(d)

Zd.

Tara noticed that the tape she was using to wrap packages was 2 centimeters wide. How many meters

wide was the tape?

Saxon Math 8/7

17.

A at

drawn on a coordinate plane with the origin. The circle intersects the x-axis at circle is

its

cente

(5, 0) anc

W

(-5, 0).

18. (3)

(a)

At what coordinates does the

(b)

What

is

circle intersect the y-axisi

the diameter of the circle?

On one tray of a balanced scale was a 50-g mass. On the other tray were

£

50

10

a small cube, a 10-g mass, and a 5-g mass. What was the mass of the

cube? Describe found your answer. small

19. l59)

how you

Paula runs begins at 27 feet below sea level and ends at 164 feet above sea level. What is the gain in elevation from the beginning to the end of the trail?

The

trail

Simplify: 20. 10 + 10 x 10

- 10

-T

10

(52)

21. 2°

2" 3

-

22. 4.5

(57)

1000 mL 1 L

L

23. 2.75 (50)

25.

4 4-

(26)

5

27. 12

17

1-

•

•

9

-

(0.8

29. 0.2

24. (23, 30)

1-

26.

8

(26)

+ 0.97)

(35)

(47)

m

+ 70

cm

(34)

( 1 l\ 3^-1^ 5^+ O 3 2

7

\^

j

6| ± [3! 3

15

-r

8

28. (2.4)(0.05)(0.005) (35)

-r

(4

x 10 2 )

30. 0.36 (45)

-r

(4

-r

0.25)

m

Lesson 62

ES

425

SON

HjX^I

Classifying Triangles

VARM-UP Facts Practice: Metric Conversions (Test

M)

Mental Math: x 8.6

a.

5

c.

lOx

e.

g.

10

+2

3

10

t*

2

b.

= 32

f.

- 4, -r number, 8

,

d.

3

2,

2 2.5 x 10"

Convert 2500 g to | of $24.00

-

x 3, + 10, V~, x 2, + 5,

4,

kg.

square that

1

Problem Solving: Marsha started the 1024-m race, ran half the distance to the finish line, and handed the baton to Greg. Greg ran half the remaining distance and handed off to Alice, who ran half the remaining distance. How far from the finish line did Alice stop? If the team continues this pattern, how many more runners will they need in order to cross the finish line?

NJEW

CONCEPT Recall from Lesson 7 that right angles,

and obtuse

we

classify angles as acute angles,

angles.

Acute angle

We use the same words to

Right angle

Obtuse angle

describe triangles that contain these

than 90°, then the triangle is an acute triangle. If the triangle contains a 90° angle, then the triangle is a right triangle. An obtuse triangle contains one angle that measures more than 90°. angles. If every angle of a triangle

measures

less

Acute

Right

Obtuse

triangle

triangle

triangle

describing triangles, we can refer to the sides and angles as "opposite" each other. For example, we might say, "The side opposite the right angle is the longest side of a right

When

426

Saxon Math 8/7

side opposite an angle_is the side the angl opens toward. In this right triangle, AB is the side opposit is the angle opposite side AB. ZC, and

The

triangle."

ZC

Each angle of

a triangle has a side that is opposite that angle

of the sides of a triangle are in the same order as the measures of their opposite angles. This means that the longest side of a triangle is opposite the largest angle, and the

The lengths

shortest side

Example

1

Name

opposite the smallest angle.

the sides of this triangle in order

from shortest

Solution

is

to longest.

note the measures of all three angles. The sum of is 59°. Since their measures is 180°, so the measure of ZH/is th e sm allest of the three angles, the side opposite ZW, which is XY, is th e shortest side. The next angle in order of size is Z X, so is the second longest side. The largest angle is ZY, so M/Xis the longest side. So the sides in order are First

we

ZW

YW

XY,YW,WX If

two angles of a

triangle are the

same measure, then

their

opposite sides are the same length.

Example 2

Which same

Solution

First

sides of this triangle are the length?

we

find that the measure of

ZQ is 61°.

So angles

Q and R

have the same measure. This means their opposite sides the

same

opposite

is

ZQ

side opposite is SR. The side SQ. So the sides that are the same length are

length.

ZR

The

are

SR and SQ. three angles of a triangle are the three sides are the same length. If all

same measure, then

all

Lesson 62

Example 3

In triangle JKL,

JK = KL =

LJ.

427

Find the

measure of ZJ.

Solution

If

two

or

more

sides of a triangle are the

same

length, then the

angles opposite those sides are equal in measure. In A JKL all three sides are the same length, so all three angles have the same measure. The angles equally share 180°. We find the measure of each angle by dividing 180° by 3.

180°

We

find that the measure of

-r

3

ZJ is

= 60° 60°.

example 3 is a regular triangle. We usually call a regular triangle an equilateral triangle. The three angles of an equilateral triangle each measure 60°, and the three sides are the same length.

The

triangle in

two sides of the same length (and thus two angles of the same measure), then the triangle is called an isosceles triangle. The triangle in example 2 is an isosceles If

a triangle has at least

triangle, as are

each of these triangles:

the three sides of a triangle are all different lengths and the angles are all different measures, then the triangle is called a

If

Here we show a scalene triangle, an isosceles triangle, and an equilateral triangle. The tick marks on the sides indicate sides of equal length, while tick marks on the arcs indicate angles of equal measure. scalene triangle.

Scalene triangles have three sides that are all different

lengths.

Isosceles triangles have at least that are the same length.

two sides

Equilateral triangles have three sides that are the same length. Equilateral triangles are regular triangles.

428

Saxon Math 8/7

Example 4

The perimeter inches long

Solution

is

of an equilateral triangle

5

Solution

How many

each side?

All three sides of an equilateral triangle are equal in length. Since 2 feet equals 24 inches, we divide 24 inches by 3 and find that the length of each side

Example

is 2 feet.

Draw an

isosceles right triangle.

Isosceles

means

same

We

is

8 inches.

the triangle has at least

two sides

that are the

means the triangle contains a right angle. right angle, making both segments equal in

length. Right

sketch a

length.

Then we complete

the triangle.

LESSON PRACTICE Practice set

Classify each triangle

g.

its

angles:

we know that two sides of an isosceles triangle are 3 cm and 4 cm and that its perimeter is not 10 cm, then what is If

its

h.

by

perimeter?

Name

the angles of this triangle in order from smallest to largest.

l

A 21

mm

20

mm

429

Lesson 62

VIIXED PRACTICE

Problem set

1. 1281

At 1:30 p.m. Dante found a parking meter that still had 10 minutes until it expired. He put 2 dimes into the meter and went to his meeting. If 5 cents buys 15 minutes of parking time, at what time will the meter expire?

Use the information in the paragraph below problems 2 and 3.

The Barkers started of gas

They

to

answer

their trip with a full tank

and

a total of 39,872 miles on their car. stopped and filled the gas tank 4 hours

later with 8.0 gallons of gas. At that time the car's total mileage was 40,060. 2.

How far

did they travel in 4 hours?

(12)

3. (46}

4. (41)

5.

The

Barkers' car traveled an average of

per gallon during the

When Use

6. -

is

a ratio

box

to solve this

Bolsheviks in the crowd.

was

Diagram

9 to 8,

this

If

is

288.

What

is

problem. There were 144 the ratio of Bolsheviks to

how many

statement.

miles

4 hours of the trip?

multiplied by w, the product the quotient when 24 is divided by w?

24

czarists

(22 48)

first

how many

czarists

were in the crowd?

Then answer

the questions

that follow.

Exit polls

showed

that 7 out of 10 voters cast

their ballot for the incumbent. (a)

According

cast their ballot for (b)

According

what percent the incumbent?

to the exit polls,

to the exit polls,

did not cast their ballot for 7.

of the voters

what fraction of the voters the incumbent?

Write an equation to solve this problem:

(60)

What number 8.

What

is

is

| of 3|?

the total price of a $10,000 car plus

8.5%

sales tax?

(46)

9. l51)

5 Write 1.86 x 10 in standard form. Then use words to

write the number.

Saxon Math 8/7

10.

Compare:

quart

1

O

1 liter

(32)

11.

Show

problem on a number

this addition

line:

(59)

(-3) + (44) + f— 9l

12.

Complete the

table.

Decimal

Fraction

Percent

(48)

5

x +

- y

y

(52)

14.

—

275%

(d)

(c)

13. Evaluate:

(b)

(a)

8

if

x = 12 and y

= 3

Find each missing exponent:

(47, 57)

(a)

2

(c)

2

5 •

2

3 -r

= 2D

3

3

2

=

15. In the figure

2°

below, angle

(b)

2

(d)

2

5 -r

2

-r

2

3

3

= 2

5

=

ZWX measures 90

D

2P c

(62)

*

Y

(a)

Which

triangle

(b)

Which

triangle is

an obtuse triangle?

(c)

Which

triangle

a right triangle?

is

is

an acute triangle?

16. In the figure at right, 119,373

are in inches

and

all

dimensions

41

angles are

right angles. (a)

What

is

the perimeter of the

figure? (b)

What

is

17. (a) Classify

the area of the figure? this

'

(37 62} -

triangle

by

its

sides. (b)

W hat

(c)

What

'(d)

7

the measure of each acute angle of the triangle? is

is

the area of the triangle?

The longest angle?

side of this triangle

is

opposite which

431

Lesson 62

Solve: 18.

-o = 1.428

::

20.

:r

-

5"

1

22.

24.

19.

*

60

1Q

:

yd 2

ft

3f

in.

2 ft

64

in.

1

hr

rni

21.

3 8'

23,

1

25.

224

bU

26.

2- -

^_

30 = "0 ^2

27.

-

.

5^2" - 10 -

mL

L - 50

20 -

1

Dra

32

.1

30. Or.

a

3

equilateral

29. Evaluate:

x - v

ere nay

box and

=

Q 3—

[ 28.

2

o: a

if

x =

3

and show

and v =

balanced scale was

a 2 50-^ r.e

triangle

mass. c:x

What was

4

its

lines

of

432

Saxon Math 8/7

LESSON

63

Symbols of Inclusion

WARM-UP Facts Practice: Fraction-Decimal-Percent Equivalents (Test L)

Mental Math: 10" 3

b. 4 x

246

x

a.

5

c.

20

e.

{196

g-

Instead of multiplying 50 and 28, double 50, find half of 28,

15

d. f.

Convert 0.5 L to mL. | of $24. 00

and multiply those numbers.

Problem Solving:

The Smiths traveled the 60-mile road to town at 60 mph. The traffic was heavy on the return trip, and they averaged just 30 mph. What was their average speed for the round trip?

NEW CONCEPTS Parentheses, brackets,

and braces

Parentheses are called symbols of inclusion. We have used parentheses to show which operation to perform first. For example, to simplify the following expression, we add 5 and 7 before subtracting their sum from 15. 15 Brackets,

and braces,

(5

+

7)

symbols of inclusion. When an expression contains multiple symbols of inclusion, we simplify within the innermost symbols first. [

],

(

},

are also

To simplify the expression 20 - [15 -

we

(5

simplify within the parentheses

20 - [15 -

Next

we

(12)]

+

7)]

first.

simplified within parentheses

simplify within the brackets.

20 17

[3]

simplified within brackets

subtracted

Lesson 63

Example

1

Solution

-

Simplify: 50 First

we

simplified within parentheses

(5)]

50 - [25]

simplified within brackets

25

Solution

subtracted

-

Simplify: 12

-

(8

|4

Absolute value symbols

-

6|

+

2)

may

serve as symbols of inclusion. In this problem we find the absolute value of 4 - 6 as the first step of simplifying within the parentheses.

12

-

(8

-

12 -

2

+

found absolute value of 4 - 6

2)

simplified within parentheses

(8)

4 Division bar

5)]

simplify within the parentheses.

50 - [20 +

Example 2

-

[20 + (10

4 33

As we noted

subtracted in Lesson 52, a division bar can serve as a

We

simplify above and below the division bar before we divide. We follow the order of operations within the symbol of inclusion.

symbol of inclusion.

r

Example v i

->

3

Solution

.r

oSimplify: F y n

4 + 5 -x 6 - 7 rf 10 - (9 - 8)

We

simplify above and below the bar before Above the bar we multiply first. Below the bar within the parentheses first. This gives us

we divide. we simplify

4+30-7 10 -

(1)

We continue by simplifying above and below the division bar. 27 9

Now we

Some

O©@0 G3 CD CD CD QSSQ CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD CD

with

divide and get

calculators with parenthesis keys are capable of dealing many levels of parentheses (parentheses within

parentheses within parentheses). When performing calculations such as the one in example 1, we press the "open parenthesis" key,

B,

for

each opening parenthesis, bracket, or brace.

We

434

Saxon Math 8/7

press

the

"close

parenthesis"

SB,

key,

each closing

for

problem in example

parenthesis, bracket, or brace. For the the keystrokes are

1,

calculations such as the one in example 3 using a calculator, we follow one of these two procedures:

To perform

1.

We perform the calculations above the bar and record the result.

2.

Then we perform

and record the result. Finally, we perform the division using the two recorded numbers. To perform the calculation with one uninterrupted sequence of keystrokes, we picture the problem like this: 4 + [10

We

below the bar

the calculations

5x6-7 -

(9

-

8)]

equals key after the 7 to complete the calculations above the bar. Then we press 83 for the division bar. We place all the operations below the division bar within a set of parentheses so that the denominator is handled by the calculator as though it were one number. press

you have

If

the

and

a calculator with parenthesis keys

algebraic

perform these calculations and note the display indicated location in the sequence of keystrokes. logic,

What number

at the

is

displayed and what does this number

is

displayed and what does this number

represent?

What number represent?

LESSON PRACTICE Practice set

Simplify: a.

30 - [40 - (10 10 +

C *

e.

2)]

b.

9-8-7

6-5-4-3

+ 2

12 + 3(8 - |-2|)

100 - 3[2(6 1

'

+ 2(3 +

4)

-

10-9(8-7)

2)]

5

4 35

Lesson 63

MIXED PRACTICE Problem set

1. (28)

and Jason each earn $6 per hour doing yard work. On one job Jennifer worked 3 hours, and Jason worked l\ hours. Altogether, how much money were

Jennifer

they paid? 2. (28.53)

3. (54)

4. (55)

When

Jim is resting, his heart beats 70 times per minute. YVhen Jim is jogging, his heart beats 150 times per minute. During a half hour of jogging, Jim's heart beats how many more times than it would if he were resting?

Use a

ratio

train

6. (51>

7. (22- 48}

solve this problem. fossil find

The was

ratio

of

2 to 9. If

many brachiopods were

During the first 5 days of the journey, the wagon train averaged 18 miles per day. During the next 2 days the train traveled 16 miles

journey

5.

to

brachiopods to trilobites in the 720 trilobites were found, how found?

wagon

(59)

box

is

have

1017 miles,

and 21 miles.

how much

If

the total

farther does the

wagon

to travel?

During one day of the journey, the wagon train descended from an elevation of 2850 feet to a spot on the desert floor 160 feet below sea level. What was the net descent of the wagon train during the day? average distance from Earth to the Sun 8 1.496 x 10 km. Use words to write that distance.

The

Diagram

this

statement.

Then answer

is

the questions

that follow.

Twelve of the 40 cars pulled by the locomotive were tankers.

8. (57)

9.

(a)

What

(b)

What percent

fraction of the cars

of the cars

were tankers? were not tankers?

3 The top speed of Jamaal's pet snail is 2 x 10~ mile per hour. Use words to write the snail's top speed.

Use

a unit multiplier to convert 1.5

km to

m.

(50)

10. Divide 4.36 l45)

by 0.012 and write the answer with a bar over

the repetend.

Saxon Math 8/7

11.

Show this

addition problem on a

number

line:

(59)

+ (+5) + (-2)

(-3)

12.

Complete the

table.

Decimal

Fraction

Percent

(48)

33%

(b)

(a) 1

(d)

(c)

3

13. Describe the rule of this function.

In

(58)

3

12 6

(36>

What

is

Out 1

— — — — — — — NO — C

T I

15

14.

F

U N

the probability of drawing a red face card

4 2 5

when

drawing one card from a normal deck?

AB ABC

In the figure below,

=

measure of angle problems 15-17.

is

15. (a) Classify

ABCDby its

AD

BD

= CD = 5 cm. The 90°. Refer to the figure for =

sides.

(62)

16.

(b)

What

(c)

Which

is

the perimeter of the equilateral triangle?

triangle is a right triangle?

Find the measure of each of the following angles:

(40)

17. (36)

(a)

ABAC

(b)

ZADB

(c)

ZBDC

(d)

ZDBA

(e)

ZDBC

(f)

ZDCB

the ratio of the length of the shortest side of A ABC to the length of the longest side?

What

is

1

Lesson 63

437

Solve: 18. (30)

— 18

—

= x +

19. 2 =

12

OAp

(45)

Simplify: 20. 3[24

-

6 + 4

+

(8

3

•

2)]

(63)

|

-1

21. 3

3

Jo22 + 4A 2 - V3

-2 22.

;

>

f56j

r52j

20mi

23 f

Igal 24>

.

50;

25.

4 qt

1 gal

14

12- -42

(26)

28. (58. 62)

1

3-

26.

3

(26. 30)

5

27. Evaluate:

(30)

x 2 + 2xy + y 2

Draw an isosceles triangle S 10W ^ s ^ ne £ symmetry.

if

1

week

-

days 7 hr 5 days 9 hr 2

5

42 + 35 + 2 3 6 9 1 6- -

3

x =

that

is

( 2

1—

I 3

3

-r

3

\ J

and y = 4

not equilateral, and

]

29. (61 >

The coordinates

of the four vertices of a parallelogram are

(0,0), (4,0), (1,-3),

and

(-3, -3).

(a)

Graph the parallelogram.

(b)

Find the area of the parallelogram.

(c)

What

is

the measure of each acute angle of the

parallelogram? 30. (3}

Three identical boxes are balanced on one side of a scale by a- 750-g mass on the other side of the scale. What is the mass of each box?

438

Saxon Math 8/7

LESSON

64

Adding Signed Numbers

WARM-UP -

Facts Practice: +

x

-r

Mixed Numbers

(Test N)

Mental Math: 2

a.

3.6 x 50

b.

7.5 x 10

c.

4x -

d.

Convert 20

e.

g.

= 35

5

V9 + 16 1.5 + 1, x

f.

2,

+

3,

-r

4,

-

cm

to

mm.

| of $1.80

1.5

Problem Solving:

When all the cards from a 52-card deck are dealt to three players, each player receives 17 cards, and there is one extra card. Dean invented a new deck of cards so that any number of players up to 6 can play and there will be no extra cards. How many cards are in Dean's deck if the number is less than 100?

NEW CONCEPT From our practice on the number line, we have seen that when we add two negative numbers, the sum is a negative number. When we add two positive numbers, the sum is a positive

number.

-3

+3

-2

-4

5)

(-2)

We have negative

-3

-2

+2

12

-1

+ (-3) = -5

3

(+2) + (+3)

4

= +5

when we add a positive number and a number, the sum is positive, negative, or zero

also seen that

depending upon which, greater absolute value.

if

either, of the

numbers has

the

4 39

Lesson 64

+5 i

!

—

•

]

TT—

I

1

i

@

i

1

1

1

-1

(+3)

1

+

(-5)

(-

1

2

-3-2-10

3

= -2

(-3)

1

(2)

+ (+5) = +2

-3 +3

©12 (+3)

We

+

(-3)

3

=

can summarize these observations with the following

statements.

1.

The sum

of

two numbers with the same sign has an

absolute value equal to the sum of their absolute values. Its sign is the same as the sign of the

numbers. 2.

The sum

3.

absolute value equal to the difference of their absolute values. Its sign is the same as the sign of the number with the greater absolute value. The sum of two opposites is zero.

of

two numbers with opposite signs has an

We can use these observations to help us without drawing a number

Example

1

line.

Find each sum: (a)

Solution

add signed numbers

(a)

(-54) + (-78)

(b)

(+45) + (-67)

(c)

(-92) + (+92)

Since the signs are the same, we add the absolute values and use the same sign for the sum. (-54) + (-78) = -132

(b)

Since the signs are different, we find the difference of the absolute values and keep the sign of -67 because its absolute value, 67, is greater than 45. (+45) + (-67) = -22

(c)

The sum no

of

two opposites

is

zero, a

sign.

(-92) + (+92) =

number which has

Saxon Math 8/7

Example 2 Solution

Find the sum:

(-3)

+

(-2)

We will show two methods. Method

Adding in order from left to right, add the first two numbers. Then add the third number. Then add the fourth number.

1:

(_5)

+ (+7) + (_4)

added -3 and -2

+ (-4)

added -5 and +7

-2

added +2 and -4

(+2)

Method

Employing

2: .

and associative terms and add all

commutative

the

rearrange the numbers with the same sign properties,

first.

+ (-2) + (-4) + (+7)

(-3)

(-9)

rearranged

+ (+7)

added

-2

added

Find each sum: (a)

+ (-3^1 f-2^1 2

(b)

These numbers are not these signed

(+4.3)

+ (-7.24)

3

V

Solution

problem

+ (-2) + (+7) + (-4)

(-3)

Example 3

+ (+7) + (-4)

numbers

is

method the method

integers, but the

the

same

as

for

for

adding adding

integers. (a)

The add the

We

^-

2-

and keep

2

6

+3-

3-

signs are both negative.

the absolute values

same

sign.

6

3

H) (b)

The

+

f

5

V

6

signs are different.

We

5g %1

find the

difference of the absolute values and keep the sign of -7.24.

-

4.3

2.94 (+4.3)

Example 4

On

+ (-7.24) = -2.94

one stock trade Tim lost $450. On a second trade Tim gained $800. What was the net result of the two trades?

441

Lesson 64

Solution

A

may be

represented by a negative number and a gain by a positive number. So the results of the two trades may be expressed this way: loss

(-450) + (+800)

The sum represents the net result of the two is +350, which represents a gain of $350.

trades.

The sum

LESSON PRACTICE Practice set*

Find each sum: a.

(-56) + (+96)

c.

(-5)

e.

(-12) + (-9) + (+16)

i.

b.

+ (+7) + (+9) + (-3)

d. (-3)

f.

(

^

V

6J

On

three separate stock trades

+

H)

(-28) + (-145)

+ (-8) + (+15)

(+12) + (-18) + (+6)

h. (-1.6)

+ (-11.47)

Dawn

gained $250, lost $300, and gained $525. Write an expression that shows the results of each trade. Then find the net result of the trades.

MIXED PRACTICE Problem set

1. l51>

2. 28)

Two

trillion is

billion? Write the

The

3.

answer in

than seven hundred

fifty

scientific notation.

taxi cost $2.25 for the first mile plus

150

additional tenth of a mile. For a 5.2-mile trip, $10 and told the driver to keep the change as a

much was

44)

how much more

each Eric paid for

tip.

How

the driver's tip?

Mae-Ying wanted to buy packages of crackers and cheese from the vending machine. Each package cost 35(2. MaeYing had 5 quarters, 3 dimes, and 2 nickels. How many packages of crackers and cheese could she buy?

Saxon Math 8/7

4. 121 '

5.

The two prime numbers p and Their difference

What

is

the

is 6.

mean

What

m

are

60.

sum?

their

is

between 50 and

of 1.74, 2.8, 3.4, 0.96, 2,

and 1.22?

(28)

6. i22 48} -

Diagram

Then answer

statement.

this

the questions

that follow.

The viceroy conscripted two

fifths of the

1200

serfs in the province.

(a)

How many

of

the

serfs

in

province

the

were

conscripted? (b)

What percent

of the serfs in the province

were not

conscripted?

7.

Write an equation to solve this problem:

(60)

What number

8.

(a)

The temperature

is

| of 100?

center of the sun is about 1.6 x 10 degrees Celsius. Use words to write that temperature. at the

7

(b)

A red blood cell is about Use words

9.

6 7 x 10~

meter in diameter.

to write that length.

Compare:

(57)

(a)

1.6 x 10

(b)

6 7 x 10"

(c)

2" 3

10. Divide

O 2"

7

O

7 x

10~ 6

O0 2

456 by 28 and write the answer

(44)

mixed number.

(a)

as a

(b)

as a decimal

(c)

rounded

rounded

to

to the nearest

two decimal

places.

whole number.

Lesson 64

11.

443

Find each sum:

(64)

12. (64>

13. dnv.

2,

62)

(a)

(-63) + (-14)

(b)

(-16) + (+20) + (-32)

On two

separate stock trades Josefina lost $327 and gained $280. What was the net result of the two trades?

The

shows an

figure

equilateral

triangle inscribed in a circle. (a)

What

is

the

measure of the

inscribed angle (b)

BCA?

Select a chord of this circle,

chord circle

14. Evaluate:

is

and

state

whether the

longer or shorter than the diameter of the

and why.

x + xy

if

x =

\

and y =

(52)

Refer to the hexagon below to answer problems 15 and 16. Dimensions are in meters. All angles are right angles.

15.

What

is

the perimeter of the hexagon?

What

is

the area of the hexagon?

(19)

16. (37)

17. 1411

18. (inv. 3)

The product the product of x and y?

The product 48.

What

is

of

x and 12

is 84.

The center

which A.

B. (-2, 1)

(4, 4)

= y 9

and 12

of a circle with a radius of three units of these points is on the circle? C. (-4, 1)

Solve:

(15)

of y

2

9

20. (44)

25x = 10

D.

is

is (1, 1).

(3, 0)

444

Saxon Math 8/7

Simplify: 21. (63)

23.

3

2

^2

+ 42 + 4

22

2

.

2

5

f26j

100 - [20 + 5(4) + 3(2 +

±+L

2l

+

2,

V

4°)]

(57, 63)

oz

25.

9oz

26 30>

5 gal 2 qt 1 pt 7

24.

+

1 gal 1 qt 1 pt

26. 0.1

-

(0.01

-

28. Write 3^b as a

(

x

if 2

4

"

'

27. 5.1

0.001)

I

-

-r

(5.1

-f-

decimal number, and subtract

it

H 1.5)

from 4.375.

(43)

29. l36)

What

is

the probability of rolling an even prime

number

with one toss of a die?

30. Figure

ABCD is

a parallelogram.

Find the measure of

(61)

B

A 58°

D (a)

ZB.

(b)

C

ZBCD.

M (c)

ZBCM.

Lesson 65

445

LESSON

n

1

65

Ratio Problems Involving Totals

Facts Practice: Metric Conversions (Test

M)

Mental Math: a.

0.42 x 50

b.

1 1.25 x 10"

c

w = if

°*.

Convert 0.75

*

CO— g.

What

1U is

m to mm.

OI 4>*i.UU

I.

the total cost of a $20.00 item plus

7%

sales tax?

Problem Solving:

Copy

this

problem and

fill

91^

in the missing digits:

NEW CONCEPT problems require that we use the problem. Consider the following problem:

Some

ratio

The

ratio of

boys

to girls at the

total to solve the

assembly was

5 to 4. If there were 180 students in the assembly, how many girls were there?

We begin by making a ratio box. for the total

number

This time

we add

a third

row

of students.

Ratio

Actual Count

5

B

4

G

9

180

boys and 4 for girls, then added these to get 9 for the total ratio number. We were given 180 as the actual count of students. This is a total. We can use two rows from this table to write a proportion. Since we were asked to find the number of girls, we will use the "girls" row.

In the ratio

column we wrote

5 for

446

Saxon Math 8/7

numbers, we will also use the row. Using these numbers, we solve the proportion.

Because "total"

we know both

total

Ratio

Actual Count

Boys

5

B

Girls

4

G

4

Total

9

180

9

180

9G = 720 G = 80

We

find there were 80 complete the ratio box.

Example

The

Solution

can use this answer

Ratio

Actual Count

Boys

5

100

Girls

4

80

Total

9

180

to

room was room totaled

ratio of football players to soccer players in the

5 to 7. If the football

48,

We

girls.

how many were

and soccer players in the

football players?

We

use the information in the problem to form a table. We include a row for the total number of players. The total ratio

number

is 12.

Ratio

Actual Count

Football players

5

F

Soccer players

7

S

12

48

Total players

_5_

F_

12

48

12F

240

F To

number

we

20

write a proportion from the "football players" row and the "total players" row. We solve the proportion to find that there were 20 football players in the room. From this information we can complete the ratio box. find the

of football players,

Ratio

Actual Count

Football players

5

20

Soccer players

7

28

Total players

12

48

Lesson 65

447

LESSON PRACTICE Practice set

Solve these problems. Begin by drawing a ratio box. a.

Acrobats and clowns converged on the center ring in the ratio of 3 to 5. If a total of 72 acrobats and clowns performed in the center ring, how many were clowns?

b.

The

young men to young women at the prom was 240 young men were in attendance, how many

ratio of

8 to 9.

If

young people attended

in all?

MIXED PRACTICE Problem set

1.

(a)

pounds of apples cost $2.40, then what is the price per pound?

(b)

what

(a)

Simplify and compare:

If 5

(46)

2.

is

the cost for 8

pounds

(41)

(b)

3. (65)

+

(0.3)(0.5)

What property

is

illustrated

a ratio

box

4.

The

5.

6.

by

this

how many

comparison?

The

ratio of big fish

to 11. If there

fish

were 1320

fish

were there?

on 15 gallons of gasoline. The

car traveled 350 miles

answer

(28)

pond was 4

how many big

car averaged

0.3(0.4 + 0.5)

to solve this problem.

to little fish in the

in the pond,

(44, 46)

O

(0.3)(0.4)

Use

of apples?

miles per gallon?

Round

the

to the nearest tenth.

The average

and 4 and 4?

of 2

reciprocals of 2

is 3.

What

is

the average of the

Write 12 billion in scientific notation.

(51)

7. (22, 36)

Diagram

this

statement.

Then answer

the questions

that follow.

One

sixth of the five

dozen eggs were cracked.

(a)

How many eggs were not cracked?

(b)

What was that

(c)

the ratio of eggs that were cracked to eggs

were not cracked?

What percent

of the eggs

were cracked?

448

Saxon Math 8/7

8.

(a)

Draw segment AB. Draw segment DC parallel to segment AB but not the same length. Draw segments between the endpoints of segments form a quadrilateral.

(b)

9. _ 2)> and

^

(

of square

EFGH are

vertices of square

^

2j

(2, 0), (0,

ABCD are

(2, 2),

coordinates of the vertices -2), (-2, 0),

and

(0, 2).

Draw

both squares on the same coordinate plane and answer (a)-(d).

(b)

What is the area of square ABCD? What is the length of one side of square ABCD?

(c)

Counting two half squares on the grid as one square

(a)

unit, (d)

what

is

the area of square

Remembering

EFGH?

that the length of the side of a square is the square root of its area, what is the length of one side of square EFGH?

5

4 57

Lesson 66

Solve: 19. (39)

™

12

1

n

.

on 20. 00)

11 — 24

+

w

11 — 12

=

Simplify: 21. 2

1

-

-

2°

2

_1

(57)

22.

4 lb 12 oz + 1 lb 7 oz

24. 16

-r

23. rsoj

f 0.04)

(0.8

3

1

12

ft

yd

in.

1 ft

-

25. 0.4[0.5

(0.6)(0.7)]

(45)

2

3

26. (26)

| O

•

if

•

4

27. 30 3

3

28. Write a

-

5 [4

+

(3)(2)

-

5]

(63)

word problem

* 12

for this division: S2.88

(W 29. (3)

Two

identical

boxes

9-ounce weight. What of each box?

30. Refer to the circle v. 2,

is

balance a the weight

with center

at

point




^

M to answer

62)

B

(a)

Name two

(b)

Classify

AAMB by its

(c)

What

the measure of inscribed angle

is

chords that are not diameters. sides.

ABC?

(a)-(c).

458

Saxon Math 8/7

I

LESSON

67

Geometric Solids

WARM-UP Facts Practice: Metric Conversions (Test

M)

Mental Math: a.

43.6

C

10

5 *

- 10 2.5

~ m 3

10

b.

3 3.85 x 10

d.

Convert 20

75%

m to decimeters (dm)

of 24

e.

10

g.

A mental calculation technique for multiplying

-f

f.

double one factor and halve the other The product is the same. Use this technique to multiply 45 and 16. is

to

factor.

x2

35 ^2 x 14

490

>

70 7

490

Problem Solving:

Tom

reads 5 pages in 4 minutes, and Jerry reads 4 pages in 5 minutes. If they both begin reading 200-page books at the same time, then Tom will finish how many minutes before Jerry?

NEW CONCEPT Geometric solids are shapes that take up space. Below we show a few geometric solids.

Sphere

Cylinder

Cone

Cube

Triangular

Pyramid

prism

Polyhedrons

Some geometric

such as spheres, cylinders, and cones, have one or more curved surfaces. If a solid has only flat surfaces that are polygons, the solid is called a polyhedron. Cubes, triangular prisms, and pyramids are examples of polyhedrons.

When

solids,

describing a polyhedron, we edges, or vertices. A face is one of the

may

refer to its faces,

flat surfaces.

formed where two faces meet. A vertex formed where three or more edges meet.

An edge is

(plural, vertices)

is

Lesson 67

459

A

prism is a special kind of polyhedron. A prism has a polygon of a constant size "running through" the prism that appears at opposite faces of the prism and determines the

name

of the prism. For example, the opposite faces of this

prism are congruent

triangles; thus this

prism

is

called a

triangular prism.

Notice that if we cut through this triangular prism perpendicular to the base, we would see the same size triangle at the cut.

we draw two identical and parallel polygons, shown below. Then we draw segments connecting

To draw as

a prism,

corresponding vertices. hidden from view.

Rectangular prism:

Then we connect

We

We

use dashes to indicate edges

draw two congruent

rectangles.

the corresponding vertices (using dashes for

hidden edges).

Triangular prism:

We

draw two congruent

We connect corresponding vertices.

triangles.

460

Saxon Math 8/7

Example

1

Use the name of a geometric solid

to describe the shape

each object:

Solution

Example 2 Solution

Example

3

Solution

(a)

basketball

(a)

sphere

(b)

rectangular prism

(c)

cylinder

A

cube has

(a)

how many

6 faces

Draw

A

(b)

shoe box

(c)

(b)

and

edges,

(a) faces, (b)

12 edges

(c)

can of beans

(c)

vertices?

8 vertices

a cube.

cube

a special kind of rectangular

is

prism. All faces are squares.

Workers involved in the manufacturing of packaging materials make boxes and other containers out of flat sheets of cardboard or sheet metal. If we cut apart a cereal box and unfold it, we see the six rectangles that form the faces of the box.

top

back

1

Q)

i

!2

front

w

1

side

bottom

we

find the area of each rectangle and add those areas, can calculate the surface area of the cereal box. If

Example 4

Which A.

of these patterns will not fold to form a cube? I

I

B.

I

1

C.

! I

1

D.

we

461

Lesson 67

Solution

Example

5

Pattern

If

D

will not fold into a cube.

each edge of a cube

the surface area (the

is 5

cm, what

is

combined area of

of the faces) of the cube?

all

5

Solution

A is

cm

cube has six congruent square faces. Each face of this cube 2 5 cm by 5 cm. So the area of one face is 25 cm and the ,

area of

all six

faces

is

6 x 25

cm 2

= 150

cm 2

lESSON PRACTICE Practice set

Use the name of a geometric solid

to describe

b.

a.

each shape:

c.

Box Tent Funnel

A triangular prism has how many of each of the d.

Faces

Draw

a

e.

g.

f.

h.

Cylinder

j.

What

to

the

are 3

cm

of

Sphere

i.

Vertices

each shape. (Refer the beginning of this lesson.)

representation

representations at

Edges

following?

three-dimensional

Rectangular prism

figure

could be formed by folding this pattern?

k.

Calculate the surface area of a cube long.

whose edges

462

Saxon Math 8/7

MIXED PRACTICE Problem set

1. (36)

The bag contains 20 red marbles, 30 white marbles, and 40 bi ue marD les. (a)

What

is

the ratio of red to blue marbles?

(b)

What

is

the ratio of white to red marbles?

(c)

If

one marble

is

drawn from the

bag,

what

the

is

probability that the marble will not be white?

2. 1303

3. (12,23)

4. (28,55)

5. (65)

6. (22 48) -

When

the product of | and \ is subtracted from the of | and |, what is the difference?

With the baby in his arms, Papa weighed 180 pounds. withcmt the baby in his arms, Papa weighed 165| pounds. How much did the baby weigh?

On 3

his first 5 tests,

t es j- s>

Che averaged 92

c n £ scored 94

points.

points, 85 points,

On

and 85

(a)

What was

his average for his last 3 tests?

(b)

What was

his average for all 8 tests?

his next points.

Use a ratio box to solve this problem. The jeweler's tray was filled with diamonds and rubies in the ratio of 5 to 2. If 210 gems filled the tray, how many were diamonds?

Diagram

this

statement.

Then answer

the questions

that follow.

Four fifths of the 360 November. (a)

How many

(b)

What percent

of the dolls

of the

dolls were sold during

were sold during November? dolls

were not sold during

November? 7.

sum

The three-dimensional figure can be formed by folding pattern has

how many

(a)

edges?

(b)

faces?

(c)

vertices?

that this

Lesson 67

8. (58,62)

Refer to the triangles below to answer (a)-(d). Dimensions meters>

^

15

6

(a)

What

is

the area of the scalene triangle?

(b)

What

is

the perimeter of the isosceles triangle?

(c)

If

one acute angle of the

right triangle

then the other acute angle measures (d)

9. (28, 34)

463

Which

What on

is

of the

two

measures

how many

37°,

degrees?

triangles is not symmetrical?

two numbers marked by arrows below?

the average of the

numb er me }

I

+

1

I

I

i

I

I

I

I

I

I

I

I

I

I

I

7.6

I

I

7.7

7.8

» 7.9

10. Write twenty-five ten-thousandths in scientific notation. (57)

Write equations to solve problems 11 and 12. 11.

What number

is

24 percent of 75?

What number

is

120%

(60)

12.

of 12?

(60)

13.

m

Find each sum: (a)

14.

(_ 2)

+ (_ 3 ) + (_ 4 )

Complete the

table.

(48)

(b)

Fraction

7 8

Use a unit multiplier

Decimal

+ (+4)

Percent

4%

(b)

(a)

15.

(+2) + (-3)

to convert

(c)

700

(d)

mm to cm.

(50)

16. In three separate stock trades Dale lost $560, gained $850, (64) and lost $280. What was the net result of the three

trades?

464

Saxon Math 8/7

17. Describe the rule of the function,

m

In

and find the missing number.

Out

x? r

U

_

49

N

— — —N L.

T

11

77

O

1

18.

-

I

Round 7856.427

(33>

(a)

to the nearest

(b) to

19. (66)

20.

hundredth.

the nearest hundred.

The diameter

of Debby's bicycle tire

24 inches. to the nearest inch?

the circumference of the tire

Consider angles A, B,

C,

is

D below.

and

(40)

120 c

B

D

(a)

Which two

angles are complementary?

(b)

Which two

angles are supplementary?

21. (a)

Show two ways

to simplify 2(5

Which property

is

illustrated in (a)?

+

(3

ft

+ 3

ft).

(41)

(b)

—W — =

22. Solve:

12

(39)

Simplify: 23. 9 + 8{7

6

•

-

5 [4

-

2

•

1)]}

(63)

24. 1

yd -

1

ft

3 in.

(56)

25. 6.4

-

(0.6

-

(35)

27. 1- + 3 (30)

29.

3-

(26)

4

3- - 14

V

+ 0.6

3

0.04)

6

6J

26. (52)

(3X0.6)

28.

- - 3-

(26,59)

5

30.

5

(52, 57)

5

2

•

5-

•

3

- V4 2 +

2~ 2

-1

What

Lesson 68

4 65

LESSON

68

Algebraic Addition

WARM-UP Facts Practice: +

-x

-f

Mixed Numbers

(Test N)

Mental Math: 0.75 + 0.5

b.

VT -

d.

12 x 2.5 (halve, double)

e.

4w - 1 = 35 20 dm to cm

g.

Find the perimeter and area of a rectangle that

a. c.

1.5

f.

33|%

of 24 is

2

m long and

m wide.

Problem Solving:

How many

different triangles are in this figure?

NEW CONCEPT Recall that the graphs of -3 and 3 are the same distance from zero on the number line. The graphs are on the opposite sides of zero.

-4

This

is

-3

-2

-1

why we say that 3 and -3

are the opposites of each other.

3 is the opposite of

-3

is

-3

the opposite of 3

as "the opposite of 3." Furthermore, -(-3) can be read as "the opposite of the opposite of 3." This means that -(-3) is another way to write 3.

We can read -3

Saxon Math 8/7

7-3. The first way is to let the minus sign signify subtraction. When we There are two ways subtract 3 from

7,

to simplify the expression

the answer

is 4.

7-3 The second way addition. To use

mean as

that -3

is

is to

= 4

use the thought process of algebraic

we let the minus sign number and we treat the problem

algebraic addition,

a negative

an addition problem. +

7

Notice that

we

difference

is

in the

We

also

can

get the

same answer both ways. The only

way we

use

= 4

(-3)

think about the problem. addition

algebraic

to

simplify

this

expression: 7

-

(-3)

We use an addition thought and think that This

is

what we

But the opposite of -3

to -(-3).

+ [-(-3)]

is 3,

7

will

added

think: 7

We

7 is

so

+

we can

[3]

write

= 10

practice using the thought process

of algebraic

addition because algebraic addition can be used to simplify expressions that would be very difficult to simplify if we used the thought process of subtraction.

Example

1

Solution

Simplify: -3

-

(-2)

We think addition. We think we are to add -3 is

what we

-(-2). This

think: (-3)

The opposite

of -2

Simplify: -(-2)

-

+ [-(-2)]

is 2 itself.

(-3)

Example 2

and

5

-

(+6)

+

So [2]

we have = -1

ai

Lesson 68

Solution

We see three numbers. We [-(-2)]

We

simplify the

II

+

[+2]

Note that

this

time

+ [-(+6)]

(-5)

and third numbers and

first

we

we have

think addition, so

+

(-5)

467

+

[-6]

get

= -9

write 2 as +2. Either 2 or +2

may be

used.

ESSON PRACTICE Practice set*

Use algebraic addition a. (-3) - (+2)

-

c.

(+3)

e.

(-8) + (-3)

(-3)

-

(-2)

d. (-3)

-

(+2)

(-8)

-

(+3) + (-2)

b.

(2)

-

sums:

to find these

(+2)

f.

-

(-4)

WXED PRACTICE Problem set

1. ll2)

2. (54)

The mass of the beaker and the liquid was 1037 g. The mass of the empty beaker was 350 g. What was the mass of the liquid?

Use a

box

1000

mL

500

problem. Adriana's soccer ball is covered with a pattern of pentagons and hexagons in the ratio of 3 to 5. If there are 12 pentagons, how many ratio

to solve this

hexagons are in the pattern? 3. (25,30)

4.

When

the

an j x

what

t

sum is

of \ and § is divided the quotient?

Pens were on sale 4

by the product of \

for $1.24.

1461

5.

(a)

What was

(b)

How much would

the price per pen?

100 pens cost?

Christy rode her bike 60 miles in 5 hours.

(46)

(a)

What was her

(b)

What was

average speed in miles per hour?

the average

ride each mile?

number

of minutes

it

took to

— 468

Saxon Math 8/7

6.

Sound

travels about 331 meters per

man y

(32.53)

seconds does

second in

sound

take

it

air.

to

Abut

travel a

kilometer? 7.

scores were

The following

made on

a test:

(Inv. 4)

72, 80, 84, 88, 100, 88, 76

8. (28, 34)

was made most

(a)

Which

(b)

What

is

the

median

(c)

What

is

the

mean

What on

is

score

number

j

10.

of the scores?

two numbers marked by arrov below?

line

I

\ |

1

1

i

•

|

!

i

I

I

I

|

M

I

!

M

»

I

I

1

1

1

j

|

I

M

This rectangular shape is two cubes tall and two cubes deep.

—

!

1

I i

10

9

8

9.

of the scores?

the average of the

——

1671

often?

11

/ / Z

How many

(a)

cubes were used to build this shape?

(b)

What

is

the

name

of this shape?

Find the circumference of each

circle:

(66)

Use 3.14

11. (58,62)

Leave n as

for n.

n.

The coordinates of the vertices of AABC are A (1,-1), b[-3,-1), and C(l,3). Draw the triangle and answer these questions: (a)

What type

of triangle

is

AABC classified by angles?

(b)

What type

of triangle

is

AABC classified by sides?

(c)

Triangle ABC's one line of symmetry passes through which vertex?

(d)

What

is

the measure of

(e)

What

is

the area of AABC?

ZB?

Lesson 68

4 69

12. Multiply (51)

twenty thousand by thirty thousand, and write the product in scientific notation.

Write equations to solve problems 13 and 13.

What number

is

75 percent of 400?

What number

is

150%

14.

(60)

14. (60)

of 1±? Z

15. Simplify: (68)

16. {67)

(a)

(-4)

-

(-6)

(b)

(-4)

-

(+6)

(c)

(-6)

-

(-4)

(d)

(+6)

-

(-4)

Find the surface area of a cube that has edges 4 inches long.

4

17.

Complete the

table.

Fraction

(48)

Decimal

in.

Percent

3

(a)

25

120%

(d)

(c)

18. Evaluate:

(b)

x 2 + 2xy + y 2

if

x = 4 and y =

5

(52)

19.

Use the name of a geometric

solid to describe each object:

(67)

(c)

(b)

(a)

20. In this figure parallelogram (40,61)

ABCD

-

divided by a diagonal into two congruent triangles. Angle DCA and ABAC have equal measures s

and are complementary. Find the measure of (a)

ZDCA.

(b)

ZDAC.

(c)

ZCAB.

(d)

ZABC.

(e)

ZBCA.

(f)

ZBCD.

Saxon Math 8/7 21. Write a

word problem

for this division: $3.00

$0.25

(13)

Solve: 22. (39)

- =

23. (1.5)

7±

C

2

= 15w

(35)

Simplify: 24. 1 gal

-

1 qt 1 pt 1

oz

(56)

25. 16

-r

(0.04

-r

0.8)

27

-

f30 ,5 2 ;

29. (52)

26. 10

-

[0.1

-

(0.01)(0.1)]

(63)

(45)

- T T 4 4

+ \ I 8 3 2 V5 -

2

4

28

«

f26j

4^ 2

•

3T

4

- l|

30. 3 + 6[l0 (63)

3

-

(3

•

4

-

5)]

471

Lesson 69

LESSON

69

More on

Scientific Notation

(VARM-UP Facts Practice: Metric Conversions (Test

M)

Mental Math: 4 - 1.5 * -

a.

r

~

4

20

e.

5

g.

M 3

b.

75 x 10~ 3

d.

18 x 35 (halve, double)

cm to dm

2

x

,

3,

-

3,

66|%

f. -r

8,

aT, x

7,

-

1,

-f

4,

of 24 x 10,

-

1,

f

,

t

2

Problem Solving:

On

a balanced scale are four identical cubes and a 12 -ounce weight distributed as shown. What is the weight of each cube?

JEW CONCEPT

When we

write a number in scientific notation, we usually put the decimal point just to the right of the first digit that is not zero. To write

4600 x 10 5 use two steps. First we write 4600 3 in scientific notation. In place of 4600 we write 4.6 x 10 in scientific notation,

we

.

Now we have 3 5 4.6 x 10 x 10

we change

two powers of 10 into one power of 10. We recall that 10 means the decimal point is 5 3 places to the right and that 10 means the decimal point is 5 places to the right. Since 3 places to the right and 5 places to the right is 8 places to the right, the power of 10 is 8. For the second step

the 3

8 4.6 x 10

Example

1

Solution

Write 25 x 10 First

we

5

in scientific notation.

write 25 in scientific notation. 5 1 2.5 x 10 x 10"

Then we combine the powers of 10 by remembering that 1 place to the right and 5 places to the left equals 4 places to the left. 2.5 x

10

-4

472

Saxon Math 8/7

Example 2 Solution

4 Write 0.25 x 10 in scientific notation.

First

we

write 0.25 in scientific notation. 4 1 2.5 x 10" x 10

Since

1

place to the

to the right,

left

and 4 places

to the right equals 3 placet

we can write 2.5 x

10 3

With practice you will soon be able

to

perform these

exercises mentally.

LESSON PRACTICE Practice set*

Write each number in scientific notation: 7 24 x 10~

a.

6 0.16 x 10

b.

c.

5 30 x 10

d. 0.75

e.

8 14.4 x 10

f.

x 10" 8

5 12.4 x 10~

MIXED PRACTICE Problem set

1. (Inv. 4)

The following

is

a list of scores Yori received in a diving

competition: 7.0

2. l65)

3. 1531

4.

6.5

6.5

7.4

7.0

6.5

(a)

Which

(b)

What

is

the

median

(c)

What

is

the

mean

(d)

What

is

the range of the scores?

score

6.0

was received the most

often?

of the scores?

of the scores?

Use a ratio box to solve this problem. The team won 15 games and lost the rest. If the team's win-loss ratio was 5 to 3, how many games were played? Brian

swam

4 laps in 6 minutes. At that rate, minutes will it take Brian to swim 10 laps?

Write each number in scientific notation:

(69)

_

c

(a)

5 15 x 10

(b)

5 0.15 x 10

how many

1

473

Lesson 69

5.

Refer to the following statement to answer

(a)-(c):

(36, 60)

The survey found that only 2 out of 5

Lilliputians

believe in giants. (a)

According

to

the

survey,

what

of

fraction

the

Lilliputians do not believe in giants? (b)

(c)

6. (661

7. (67)

8.

If

60 Lilliputians were selected for the survey,

many

of

What

is

them would believe

in giants?

the probability that a randomly selected Lilliputian who participated in the survey would believe in giants?

The diameter

stump was 40 cm. Find the stump to the nearest centimeter.

of the tree

circumference of the tree

Use the name of a geometric

solid to describe the shape of

these objects: (a)

volleyball

(a)

What

(58, 62)

is

(b)

water pipe

tepee

(c)

the perimeter of the

equilateral triangle at right? (b)

What

is

the measure of each of

1

1

1

1

1

1

1

1

1

1

1

1 1

1

1 1 1

its

(c)

angles?

Trace

the

paper, and

inch

triangle

show

its

on

your

lines

of

symmetry. 9.

Simplify:

(68)

10.

how

(a)

(-4) + (-5)

(b)

(-2)

-

(-6)

+ (-3) - (-4) - (+5)

Find the circumference of each

circle:

(66)

Use 3.14

for;r.

Use

—

22

for n.

1

474

Saxon Math 8/7 11. Refer to the figure to 1371

Dimensions

answer

millimeters.

in

are

(a)-(c).

Corners that look square are square. (a)

What

of

area

the

is

the

hexagon? the area of the shaded triangle?

(b)

What

is

(c)

What

fraction of the

hexagon

is

shaded?

Write equations to solve problems 12 and 13. 12.

What number

is

50 percent of 200?

What number

is

250%

(60)

13.

of 4.2?

(60)

14.

Complete the

table.

Decimal

Fraction

(48)

3

20

(b)

(a)

150%

(d)

(c)

15. Refer to this figure to

Percent

answer

(a)-(c):

(40)

—

•

Q R (a)

Which

angle

is

supplementary

(b)

Which

angle

is

complementary

(c)

If

to to

ZSPT?

ZSPT?

ZQPR measures 125°, what is the measure of ZQPT?

16. Evaluate: a

2

- Va

+ ab - a

if

a = 4 and b = 0.5

(52, 57)

17. Describe the rule of this function, 1581

F

In

and find the missing number. 8 6

10 4

— — — — — — ON — —

Out

U N

15

C

11

T I

19

4 75

Lesson 69

144 by 11 and write the answer

18. Divide

19. l41)

decimal with a bar over the repetend.

(a)

as a

(b)

rounded

Anders used

to the nearest

this

whole number.

formula to convert from degrees Celsius

to degrees Fahrenheit:

°F = 1.8°C + 32 the Celsius temperature (°C)

If

what

20°C,

is

is

the

Fahrenheit temperature (°F)? 20. (2i,28)

The prime number 19 p r j me numbers?

is

the average of which

two

different

Solve: 21.

t

+ | = if

22.

16

8

(30)

(39)

fO =

^ ^

Estimate each answer to the nearest whole number. Then perform the calculation.

(26)

2^

3-

23. V

-r

1-

24.

3

(26,30)

4

4^ + [5^

16

2

-r

l\ 3

Simplify: 25.

5

ft

7 in.

(49)

+ 6

ft

8 in.

350

m

oc Zb.

Is

(so)

27. 6

-

(0.5

60s 1 min -f

1

km

1000

m 28. $7.50

4)

29.

-r

0.075

(45)

(35)

Use prime factorization

to

reduce

|f|.

(24)

30. (a)

Convert 2\ to a decimal and add 0.15.

(43)

(b)

Convert 6.5 to a mixed number and add

|.

476

Saxon Math 8/7

LESSON

70

Volume

WARM-UP Facts Practice: +

—x

Mixed Numbers

(Test N)

Mental Math: 25

2

a.

4.8 + 3 + 0.3

b.

c.

5m -

d. $4.80 x

e.

20

g.

Find the perimeter and area of a square that has sides 0.5

3

= 27

dm to mm

f.

60%

50 (halve, double)

of 25

m long.

Problem Solving:

Copy

this

problem and

fill

in the missing digits:

679

NEW CONCEPT Recall from Lesson 67 that geometric solids are shapes that take up space. We use the word volume to describe the space

occupied by a shape. To measure volume, we use units that occupy space. The units that we use to measure volume are cubes of certain sizes. We can use sugar cubes to help us think of volume.

Example

1

Solution

This rectangular prism was constructed of sugar cubes. Its volume is how many cubes?

To

"7*

s / z

we

calculate the number of cubes it contains. We see that there are 3 layers of cubes. Each layer contains 3 rows of cubes with 4 cubes in each row, or 12 cubes. Three layers with 12 cubes in each layer means that the volume of the prism is 36 cubes.

find the

volume

of the prism,

Volumes are measured by using cubes of a standard size A cube whose edges are 1 centimeter long has a volume of 3 1 cubic centimeter, which we abbreviate by writing 1 cm .

1

cm 1

1

cm

cm

1

cubic centimeter =

1

cm'

477

Lesson 70

Similarly, 1 1

if

each of the edges is If each of the edges is

cubic foot. cubic meter. 1

To

cubic foot = the

calculate

1 ft

3

1

volume of

volume the volume

1 foot long, the 1

meter long,

cubic meter = a

1

m

is is

3

we can imagine

solid,

constructing the solid out of sugar cubes of the same size. We would begin by constructing the base and then building up the layers to the specified height.

Example 2

Find the number of 1-cm cubes that can be placed inside a rectangular box with the dimensions shown.

3

cm 4 5

Solution

box is 5 cm by 4 cm, so we can place 5 rows of 4 cubes on the base. Thus there are 20 cubes on the

The base

of the

cm

cm

y s s s s

;

first layer.

Since the box is 3 cm tall, 3 layers of cubes in the box.

20 cubes x 3 layers 1

We Example

3

can

fit

= 60 cubes

lay^f

find that 60

What

we

is

the

1-cm cubes can be placed

volume

of

this

in the box.

cube?

Dimensions are in inches.

Solution

The base

is

4

in.

by 4

in.

Thus 16 cubes

can be placed on the base. Since the big cube is 4 in. 4 layers of small cubes. 16 cubes x 4 layers

tall,

there are

= 64 cubes

si.-

X

1 layer

Each small cube has a volume of 1 cubic inch. Thus the 3 volume of the big cube is 64 cubic inches (64 in. ).

478

Saxon Math 8/7

LESSON PRACTICE Practice set

a.

b.

was prism rectangular This constructed of sugar cubes. Its volume is how many sugar cubes?

Find the number of 1-cm cubes that can be placed inside a box with dimensions as illustrated.

/ /

/ 10

7

1

cm 10

c.

cm

10

cm

volume of this rectangular prism? Dimensions are

What

the

is

in feet. 10

d.

As

a class, estimate the

meters.

Then use

volume

of the classroom in cubic

a meterstick to

measure the

length,

width, and height of the classroom to the nearest meter, and calculate the volume of the room.

MIXED PRACTICE Problem set

1. (53)

was 38 kilometers from the encampment to the castle. Milton galloped to the castle and cantered back. If the round trip took 4 hours, what was his average speed in It

kilometers per hour? 2.

The

(37, 58)

(0, (a)

two angles of a triangle are The y-axis is a line of symmetry of the

vertices of

-4).

What

(3, 1)

and

triangle.

are the coordinates of the third vertex of the

triangle? (b)

3. l66)

What

is

the area of the triangle?

Using a tape measure, Gretchen found that the circumference of the great oak was 600 cm. Using 3 in place of n, she estimated that the tree's diameter was 200 cm. Was her estimate for the diameter a little too large or a little too small?

4.

Grapes were priced

Why?

(a)

pounds for What was the price per pound?

(b)

How much would

1461

at 3

$1.29.

10 pounds of grapes cost?

Lesson 70

5. (35)

6.

479

the product of nine tenths and eight tenths is subtracted from the sum of seven tenths and six tenths, what is the difference? If

Three fourths of the

batter's

188 hits were singles.

(22, 48)

(a)

How many

(b)

What percent of the batter's

7.

On

(8)

« 1

8. 1701

an inch

1^-inch

of the batter's hits were singles?

ruler,

were not singles?

hits

which mark

is

halfway between the

mark and the 3 -inch mark?

Find the number of 1-cm cubes that can be placed in the box at

3

cm

^5. 5 cm

right. 5

9.

Find the circumference of each

cm

circle:

(66)

(b)

Leave k as

10. Write

11. (28, 34)

12 x 10

What

for k.

each number in scientific notation:

(69)

(a)

Use 3.14

n.

-6

(b)

0.12 x 10

the average of the three arrows on tne num ber line below? is

-6

numbers marked by

++++

12.

Use

1.0

0.9

0.8

0.7

a unit multiplier to convert 1.25 kilograms to grams.

(50)

13.

Find each missing exponent:

(47, 57)

(a)

2

(c)

2

6 •

3

14. Write

-r

2

3

2

= 6

=

2^ 2

n

an equation

(b)

2

(d)

2

to solve this

6 -r

6

2

3

- 26 =

problem:

(60)

What number

is

= 2D

| of 100?

2°

Saxon Math 8/7

15.

Complete the

table.

Decimal

Fraction

Percent

(48)

14%

(b)

(a) 5

(d)

(c)

6

16. Simplify: (68)

(a)

(-6)

-

(-4) + (+2)

(b)

(-5)

+

(-2)

ab -

17. Evaluate:

-

-

(-7)

(a

-

b)

(+9) if

a = 0.4 and b = 0.3

(52)

18.

29,374.65 to the nearest whole number.

Round

(42)

19. Estimate the

product of 6.085 and 7 11 16'

(29, 33)

20. (67)

21.

What three-dimensional be formed by folding

figure

can

this pattern?

Draw

the three-dimensional figure.

What

is

the surface area of a cube with edges 2

long?

ft

(67)

Solve: 22. 4.3

= x - 0.8

23. 09)

(35)

1 d

=

M 1.5

Simplify:

10

24. (56)

25.

lb

-

6 lb 7 oz

26.

3 3-

-f

(26)

4

f

2

(53)

\

1-

3

•

27.

3

28. (0.06

-r

f3oj

5)

-r

(a)

What

(b)

A 15%

1461

days

1

day

1

week

111

4^ + 5^ - l| 2

6

3

0.004

29. Write 9| as a decimal 30.

5

8 hr

$5.25 1 hr

number, and multiply

it

by

9.2.

the total price of a $15 meal including sales tax? is

tip

on a $15 meal would be

6%

how much money?

Investigation 7

481

INVESTI Focus on

Balanced Equations Since Lesson 3 we have solved equations informally by using various strategies for finding the missing number. In this investigation

we

will practice a

for solving equations.

perspective,

more

To help us

we will use

formal, algebraic

method

see equations from this

new

a balance scale as a visual aid.

Equations are sometimes called balanced equations because the two sides of the equation "balance" each other. A balance scale can be used as a model of an equation. We replace the equal sign with a balanced scale. The left and right sides of the equation are placed on the left and right trays of the balance. For example, x + 12 = 33 becomes

x + 12

Using a balance-scale model we think of how to get the unknown number, in this case x, alone on one side of the scale. Using our example, we could remove 12 (subtract 12) from the left side of the scale. However, if we did that, the scale would no longer be balanced. So we make this rule for ourselves.

Whatever operation we perform on one side of an equation, we also perform on the other side of the equation to maintain a balanced equation.

We see that there are two

steps to the process.

Step

1:

Select the operation that will isolate the variable.

Step

2:

Perform the selected operation on both sides of the equation.

482

Saxon Math 8/7

In our

example we

select "subtract 12"

as the operatioi

required to isolate x (to "get x alone"). Then operation on both sides of the equation. Select operation:

To

we perform

x + 12

33

x

21

thi;

isolate x, subtract 12.

=^^^=

Perform operation:

To keep the scale balanced, subtract 12 from both sides of the equation. After subtracting 12 from both sides of the equation, x is isolated on one side of the scale, and 21 balances x on the other side of the scale. This

solution by replacing

shows

x with 21

x + 12 = 33 21 + 12 = 33 33 = 33

illustrate a

21.

We check our

in the original equation.

original equation

replaced x with 21 simplified

Both sides of the equation equal solution, x = 21, is correct.

Now we will

x =

that

33.

left

side

This shows that the

second equation, 45 = x +

18, with

a balance-scale model.

x + 18

45

This time the unknown number balance scale, added to 18.

is

on the

right side of the

1.

Select the operation that will isolate the variable, and write that operation on your paper.

2.

Describe how to perform the operation and keep a balanced scale.

Investigation 7

3.

Describe what will remain on the left and right side of the balance scale after the operation is performed.

We show the

line-by-line solution of the equation below.

45 = x + 18

original equation

45 - 18 = x + 18 - 18

We

483

subtracted 18 from both sides

27 = x +

simplified both sides

27 = x

x +

= x

check the solution by replacing x with 27 in the original

equation.

45 = x + 18

original equation

45 = 27 + 18

replaced x with 27

45 = 45

simplified right side

the solution in the original equation, we see that the solution is correct. Now we revisit the equation to

By checking illustrate

one more

idea.

x + 18

4.

Suppose the contents of the two trays of the balance scale were switched. That is, x + 18 was moved to the left side, and 45 was moved to the right side. Would the scale still be balanced? Write what the equation would be.

Now

we

an equation consider multiplication rather than addition. will

that

involves

2x = 132 Since 2x means two x's (x + on a balance scale two ways. 2x

132

x),

we can show x + x

this equation

132

484

Saxon Math 8/7

We

must perform the operations necessary to get one x alone on one side of the scale. We do not subtract 2, because 2 is not added to x. We do not subtract x, because there is no x to subtract from the other side of the equation. To isolate x in this equation, we divide by 2. To keep the equation balanced, we divide both sides by 2.

Our goal

is

to isolate x.

Select operation:

To

isolate x, divide

2x

132

x

66

by 2.

Perform operation:

To

keep

the

equation

divide both sides by

Here

we show

C_^>

2.

the line-by-line solution of this equation.

2x = 132

original equation

2* = 132

divided both sides by y 2

2

2

lx = 66

x = 66 Next

balanced,

we show

simplified both sides

lx = x

the check of the solution.

2x = 132 2(66) = 132

132 = 132

original equation

replaced x with 66 simplified

left

side

This check shows that the solution, x = 66,

is correct.

5.

Draw

6.

Select the operation that will isolate the variable, and

a balance-scale

model

for the equation

3x = 132.

write that operation on your paper.

7.

8.

Describe how to perform the operation and keep a balanced scale.

Draw

a balance scale

and show what

the scale after the operation

is

is

on both sides

performed.

of

Investigation 7

9.

10.

485

Write the line-by-line solution of the equation.

Show

the check of the solution.

Most students choose

solve the equation 3x = 132 by

to

dividing both sides of the equation by 3. There is another operation that could be selected that is often useful, which we will describe next. First note that the number multiplying the variable, in this case 3, is called the coefficient of x. Instead of dividing by the coefficient of x, we could choose to multiply by the reciprocal of the coefficient. In this case we could multiply by |.

3x = 132 1 •

3x =

1-132

3

lx =

132 3

x = 44

When

solving equations with whole number or decimal number coefficients, it is usually easier to think about dividing by the coefficient. However, when solving equations

usually easier to multiply by the reciprocal of the coefficient. Refer to the following equation for problems 11-14:

with fractional coefficients,

it is

1X 4

- JL " 10

11. Select the operation that will result in

fx becoming lx in

the equation. 12. Describe

how

to

perform the operation and keep the

equation balanced. 13. Write a line-by-line solution of the equation.

14.

We

Show

the check of the solution.

find that the solution to the equation is § (or arithmetic we usually convert improper fractions to

l|).

In

mixed

;

486

Saxon Math 8/7

mm numbers. In algebra we usually leave improper fractions in I improper form unless the problem states or implies that a' mixed number answer is preferable. For each of the following equations,

you

(a) state

select to isolate the variable, (b) describe

the operation

how

the operation and keep the equation balanced, line-by-line solution of the equation,

and

(d)

to

perform

write a the check

(c)

show

of the solution. 15.

x +

16. 3.6

2.5

=

7

= y + 2

17.

4w

18.

1.2m = 1.32

21.

Make up an

= 132

addition equation with decimal numbers.

Solve and check 22.

Make up

it.

a multiplication equation with

coefficient. Solve

and check

it.

a fractional

'

Lesson

|L

E S S

487

71

O N

71

Finding the Whole Group When a Fraction Is Known

WARM-UP Facts Practice: Classifying Quadrilaterals and Triangles (Test O)

Mental Math: a.

(-3) + (-12)

c

loo

-

=

50

g.

What

d.

lo

cm to

e.

is

b. 4.5

m

f.

x 10" 3

12 x 2| (halve, double)

75%

of $36

the total cost of a $30 item plus

8%

sales tax?

Problem Solving: Bry has three different shirts and three different ties he can wear with each shirt. How many different shirt-tie combinations can Bry wear? If the shirts are designated A, B, and C, and the ties 1, 2, and 3, one combination is Al. List all the possible combinations.

NEW CONCEPT diagrams of fraction problems understand problems such as the following:

Drawing

can

help

us

of the fish in the pond are bluegills. If there are 45 bluegills in the pond, how many fish are in the pond?

Three

fifths

The 45

We

bluegills are 3 of the 5 parts. divide 45 by 3 and find there are

15 fish in each part. Since there are 15 fish in each of the 5 parts, there are 75 fish in all.

fish

—

were

1

(45).

15

fish

15

fish

15

fish

15

fish

were not

finished page 51, he was § of the book. His book had how many pages?

When Juan

fish

5 bluegills

bluegills.

Example

15

way through

his

|

488

1

|

Saxon Math 8/7

Solution

This is 3 of 8 parts of the book. Since 51 -r 3 is 17, each part is 17 pages. Thus the pages.

Juan read 51

totals 8 x

whole book (8 parts) which is 136 pages.

pages 1

^

read

8

•

17 pages

(51)

17,

^

not yet

<


(b)

0.1%

(d)

(c)

18.

Percent

A

square sheet of paper with an 2 area of 81 in. has a corner cut off, forming a pentagon as shown.

What

(a)

is

the perimeter of the

pentagon?

What

(b)

is

the

area

of

the

pentagon? 19. dm-. 6)

What type

of parallelogram has four congruent angles but

nQ necessar i}y four congruent sides? j.

20. {28]

When

water increases in temperature from its freezing point to its boiling point, the reading on a thermometer increases from 0°C to 100°C and from 32°F to 212°F. The temperature halfway between 0°C and 100°C is 50°C. What temperature is halfway between 32°F and 212°F?

For problems 21-24, solve and check each equation. each step. 21.

x - 25 = 96

22. (Inv. 7)

(Inv. 7)

23. 2.5p = 6.25

24.

-m

= 12

3

10 = / + 3^

(Inv. 7)

(Inv. 7)

Simplify: 25. (20)

\13' -

5'

26. 1 ton (16)

- 400

lb

Show

520

Saxon Math 8/7

27.

3-

(26, 43)

4

28. (35, 43)

x

4-

x (0.4)

2

(fraction answer)

6

1

1

8

4

3- + 6.7 + 8— (decimal answer)

29. (a) (-3) + (-5)

-

(-3)

-

|+5|

(68)

(b)

(-73) + (-24)

-

(-50)

whether the quotient is greater and state why. Then perform the

30. Before dividing, determine

than or less than

1

calculation.

5

6 2 3

Lesson 77

521

LESSON

77

Percent of a Number, Part 2

(VARM-UP Facts Practice: +

-x

-r

Integers (Test P)

Mental Math: a.

(+12)

c.

f§§

e.

25

g.

-

=

f

d.

cm to m 10% of 50,

x 10 6

b. 4

(-18)

f.

x 6,

+

2,

t

4,

x 2,

Estimate 12

f

,

is

15%

of $61.

| of n.

x 9,

f

,

x

7,

t 2

Problem Solving:

On

balanced scale are a 25-g mass, a 100-g mass, and five identical blocks marked X, which are distributed as shown. What is the mass of each block marked X? Write an equation illustrated by this balanced scale. a

JEW CONCEPT we

practiced fractional-part problems involving fractions and decimals. In this lesson we will practice similar problems involving percents. First we translate the problem into an equation; then we solve the equation.

In Lesson 74

Example

1

Solution

What percent

of 40

is

25?

We translate the question to an equation and solve. What percent

of 40

I

WP To

solve

we

25?

WW is

x 40 = 25

question

equation

divide both sides of the equation by 40.

WP**« 40

= 25 40

divided b J 4Q

l

simplified

Since the question asked "what percent" and not "what fraction," we convert the fraction | to a percent.

-

x

100% = 62^% 2

8

So 62f %

(or

62.5%) of 40

is 25.

converted to a percent

522

Saxon Math 8/7

Example

2

Solution

What percent

We translate

of $3.50

and

$0.28?

is

solve.

What percent

of $3.50

Wr

x $3.50

is

$0.28?

question

$0.28

equation

l

W

$0.28

P x $3r5fi

divided by $3.50

$3.50 l

WP We perform the

WP This

Solution

canceled

3.5

=

0.28

2.8

3.5

35

= 0.08

divided

The question asked decimal 0.08 to 8%.

a decimal answer.

is

WP 3

0.28

decimal division.

answer so we convert the

Example

=

=

8%

converted to a percent

Seventy-five percent of

what number

translate the question to

what number

solve,

we

multiply both sides by 100 over 75.

x

76 i

WN

= 600

WN

= 800

100

multiplied by ±f

75

i

simplified

We could have used the fraction 4 for what percent

Since 50

100%.

equation

i

l

Fifty is

question

= 600

N

100

600?

II

75

To

is

I

I

Solution

600?

is

an equation and solve. We can translate 75% to a fraction or to a decimal. We choose a fraction for this example.

We

Seventy-five percent of

Example 4

for a percent

is

We translate to

50

the

same

result.

answer will be greater than an equation and solve. 40, the

what percent I

=

100

of 40?

more than

Fifty is

^ with

W,

of 40? I

question

I

x 40

equation

,

Lesson 77

We

5 23

divide both sides by 40.

WP

50

£6

x

40

divided by 40

£6 l

5

WP

- =

We

convert | to a percent.

WP Example

5

Solution

Sixty

We

is

=

100% = 125%

x

|

converted to a percent

150 percent of what number?

by writing 150%

translate

fraction.

We

simplified

We will use the

as

either a decimal or a

decimal form here.

Sixty

is

150%

of

J

|

J

J

60

=

1.5

x

what number? |

WN

equation

divide both sides of the equation by l

60

question

WN

x

1.5.

divided by 1.5

1.5 l

WN

40 =

simplified

ESSON PRACTICE Practice set*

what percent

of 40?

a.

Twenty-four

b.

What percent

c.

Fifteen percent of

what number

d.

What percent

is

e.

Twenty-four

f.

is

of 6

of 4

is

is

2?

5,

of

what number?

writing

150%

as a decimal.

g.

What percent

45?

6?

120%

Rework example

is

of $5.00

is

$0.35?

as a fraction instead of

,

524

Saxon Math 8/7

MIXED PRACTICE Problem set

1. (65)

Use

a ratio

box

to

solve this problem.

Tammy

savet

nickels and pennies in a jar. The ratio of nickels t( pennies was 2 to 5. If there were 70 nickels in the jar

how many

coins were there in all?

Refer to the line graph below to answer problems 2-4. Jeremy's Test Scores 100

o

90

CD i— i—

o

2

(b)

x
.

x - y =

™ = a

•

-1

Use a case 1-case 2 ratio box to solve Four inches of snow fell in 3 hours. At that

would

l71)

1

a separate

this

problem.

rate,

how long

to fall?

Then answer

the questions

that follow.

Twelve students earned A's. This was | of the students in the class. (a)

How many

(b)

What percent

students did not earn A's? of the students did not earn A's?

Write equations to solve problems 9-12. 9.

Thirty-five

is

70%

what number?

of

(77)

10.

What percent

of 20

is

17?

What percent

of 20

is

25?

(77)

11. (77)

12. (77)

Three hundred sixty

is

75 percent of what number?

5 35

Lesson 79

13. Simplify: (73)

(a)

(c)

14.

1^4 -8

(b)

-12(12)

(d)

=1« +6

Complete the

table.

(48)

-16(-9)

Decimal

Fraction

Percent

i

(b)

(a)

25

8%

(d)

(c)

15.

At the Citrus used car lot, a salesperson is paid a commission of 5% of the sale price for every car he or she sells. If a salesperson sells a car for $4500, how much would he or she be paid as a commission?

%

simplify:

17.

A

m

(75>

iS

square sheet of paper with an area of 100 corner cut off, as shown in the figure below.

(a)

What

is

the perimeter of the shape?

(b)

What

is

the area of the shape?

18. In the figure at right, (67, 70)

^

What

is

the

volume

of this

Z

rectangular prism? (b)

19.

What

the total surface area of the rectangular prism? is

Find the circumference of each

circle:

(66)

Use 3.14

for n.

2

has a

each small cube has a volume of

cu bi c centimeter.

(a)

in.

Leave n as

n.

536

Saxon Math 8/7

each triangle as acute, right, or obtuse. Then identify each triangle as equilateral, isosceles, or scalene.

20. Identify t62)

(b)

(a)

For problems 21-23, solve and check each equation. Show each step. 1

22. 3^ 3 (Inv. 7)

= 2.88

21. 1.2x (Inv. 7)

23. (Inv. 7)

%w

5 x + -

6

= 10

2

Simplify:

100 +

3

5[3

-

2

2(3

24.

+

3)]

(63)

25. 1561

27. (26)

3

-

2

min 24 hr 45 min 30 hr 15

1 I"

29.

t"

30. (a)

(--10)

1.5

(68)

(b)

(--10)

s

26.

1 yd J

(50)

-3ft

2

•

•

1

yd

1

s

28.

9

(decimal answer)

(30)

11

5 4- + 3^ + 7-

6

3

4

yd

Lesson 80

E

S

S

537

O N

80

Transformations

VARM-UP Facts Practice: Classifying Quadrilaterals and Triangles (Test O)

Mental Math:

-

b.

40"

d.

15

a.

(-30)

c.

5q - 4 = 36

e.

1500

g.

Find the perimeter and area of a square with sides

(+45)

m to km

f.

is |

What

of n.

is

4 of 15 9 1.5

m long.

Problem Solving: terms of the following sequence. next two terms.

Lree Le

EW CONCEPT they are the same These two triangles are congruent, but they

Recall that two figures are congruent

shape and

size.

are not in the

same

if

position:

We

can use three types of position change to move triangle I to the position of triangle II. One change of position is to '"flip*" triangle I over as though flipping a transparent piece of paper.

i

n

538

Saxon Math 8/7

A second change of position is to

"slide" triangle

third change of position

"turn" triangle

is to

I

to the right.

ii

i

The

I

and

1

90° clockwise.

II

These "flips, slides, and turns" are called transformations and have special names, which are listed in this table. Transformations

A

Movement

Name

flip

reflection

slide

translation

turn

rotation

reflection of a figure in a line (a "flip") produces a mirror

image of the figure that If

of

we

reflect

A ABC

AABC appears

is

reflected.

in the y-axis, the reflection of every point

on the opposite side of the y-axis the same

distance from the y-axis as the original point. We can refer to the reflected triangle as AA'B'C which we read as "triangle A prime, B prime, C prime." '

y

,

5 39

Lesson 80

we

then reflect AA'B'C in the x-axis, we see AA"B"C" ("triangle A double prime, B double prime, C double prime") in the fourth quadrant.

If

y

Example

1

The coordinates of the vertices and T(l,l). Draw ABST and AR'S'T'.

Solution

We

What

of

ABST are B

its

(4, 3),

S

(4, 1),

reflection in the x-axis,

are the coordinates of the vertices of AB'S'T'?

ABST and draw the triangle. The of ABST in the x-axis appears on the

graph the vertices of

reflection of every point

opposite side of the x-axis the same distance from the x-axis as the original point. We locate the reflected vertices and draw AB'S'T'. y

6 5

Note that

if

a segment

reflection, the

were drawn between a point and

segment would be perpendicular

its

to the line of

the x-axis. The coordinates of the vertices of AB'S'T' are R' (4, -3), S' (4, -1), and T' (1, -1). reflection,

which

in this case

is

— 540

Saxon Math 8/7

translation "slides" a figure to a new position without turning or flipping the figure. If we translate quadrilateral JKLM 6 units to the right and 2 units down, quadrilateral

A

J'K'L'M' appears in the position shown. To perform the transformation we translate each vertex 6 units to the right and 2 units down. Then we draw the sides of the quadrilateral. y

M

J i

M

J'

I K

L

/

K

L

Example 2

The coordinates

B

(4, 1),

C(l,

1),

of the vertices of rectangle

D (1,

and

Solution

(4, 3),

Draw ttABCD and its image, left 5 units and down 4 units.

3).

\Z\A'B'C'D', translated to the

What

ABCD are A

are the coordinates of the vertices of

XHA'B'C'D' ?

We

graph the vertices of \Z\ABCD and draw the rectangle. Then we graph its image by translating each vertex 5 units to

the

left

and 4 units down. y

6 5

-4 -3

A

D

-2 1

6

u

c

-3-2

A

l

x

c

B

12

3

4

5

6

B -5 -6

We find that the coordinates A'

(-1, -1),

B'

(-1, -3),

of the vertices of HIA'B'C'D' are

C (-4, -3), and D

'

(-4, -1).

Lesson 80

541

A

rotation of a figure "turns" the figure about a specified point called the center of rotation. At the beginning of this lesson we rotated triangle I 90° clockwise. The center of

rotation

was the vertex

below, triangle

One way

of the right angle. In the illustration

ABC is rotated

180° about the origin.

view the effect of a rotation of a figure is to trace the figure on a piece of transparency film. Then place the point of a pencil on the center of rotation and turn the to

transparency film through the described rotation.

Example

3

The coordinates of the vertices of APQR are P (3, 4), Q (3, 1), and R (1, 1). Draw APQR and also draw its image, AP'Q'R', counterclockwise rotation of 90° about the origin. What are the coordinates of the vertices of AP'Q'R'? after a

Solution

We graph the vertices

Then we place

of APQR

and draw the

triangle.

of transparency film over the also place a mark coordinate plane and trace the triangle. on the transparency aligned with the x-axis. This mark will a

piece

We

align with the y-axis after the transparency

is

rotated 90°.

542

Saxon Math 8/7

After tracing the triangle on the transparency, point of a pencil on the film over the origin,

center of rotation in this

paper

we

still,

rotate

we

place the

which

is

the

example. While keeping the graph the

film

90°

(one-quarter

turn)

counterclockwise. The image of the triangle rotates to the position shown, while the original triangle remains in place.

We name

image AP'Q'R' and through

the

transparency see that the coordinates of the vertices P'(-4, 3), Q' (-1,3), and 1).

are

the

rotated

LESSON PRACTICE Practice set

a.

Perform each of the examples in this lesson not already done so.

b.

The

vertices of rectangle

Y(l,

1),

and Z(l,

dW'X'Y'Z' after origin. What are

3).

WXYZ

if

you have

W(4, 3), X(4, 1), Draw the rectangle and its image are

a 90°

clockwise rotation about the the coordinates of the vertices of

rwx'Y'z"? c.

The

AJKL are /(1,-1), K[3,-2), and L(l,-3). Draw the triangle and its image after reflection in the y-axis, AJ'K'L'. What are the coordinates of the vertices of vertices of

AJ'K'L'?

d.

The coordinates

Q (-1,

1),

R

CJPQRS are P (0, Draw CJPQRS and

of the vertices of

(-4, 1),

and S

(-3, 3).

3), its

image CJP'Q'R'S' translated 6 units to the right and 3 units down. What are the coordinates of the vertices of CJP'Q'R'S'?

543

Lesson 80

IXED PRACTICE

Problem set

1. (55)

2.

mowed

hours and earned $7.00 per hour. Then she washed windows for 3 hours and earned $6.30 per hour. What was Tina's average hourly pay for the 7-hour period?

Tina

Evaluate:

lawns

x +

(x

2

for 4

-

- y

xy)

if

x = 4 and y =

3

(52)

3.

Compare: a

Ob

ah =

if

2

(79)

4. 1651

5. 02, 66)

6.

box to solve this problem. When Nia cleaned her room, she found that the ratio of clean clothes to dirty clothes was 2 to 3. If 30 articles of clothing were discovered, how many were clean?

Use

a ratio

The diameter of a c i rcum ference f a

half-dollar

is 3

centimeters. Find the

half-dollar to the nearest millimeter.

Use a unit multiplier

to convert \\ quarts to pints.

(50)

7. (78)

8. 1721

Graph each inequality on a separate number (a) x > -2 (b) x < .

.

Use a ratio box to solve this problem. In 25 minutes, 400 customers entered the attraction. At this rate, how

many customers would 9. 1711

Diagram

enter the attraction in 1 hour?

statement.

this

Then answer

the questions

that follow.

Nathan found

that

it

joint to his hip joint.

was 18 inches from his knee This was j of his total height.

(a)

What was Nathan's

total height in

(b)

What was Nathan's

total height in feet?

inches?

Write equations to solve problems 10-13. 10. Six

hundred

is

f of

what number?

(74)

11.

Two hundred

eighty

What number

is

is

what percent

(77)

12.

4 percent of 400?

(60)

13. Sixty is (77)

line:

*

,

60 percent of what number?

of 400?

544

Saxon Math 8/7 14. Simplify: (73)

f

,

(a)

(c)

-600

600

(b)

20(-30)

(d)

15. Anil is paid a (60>

16.

-12

=15

+15(40)

commission equal Anil

to

6%

each appliance he

sells.

$850, what

commission on the

is

Complete the

Anil's

table.

If

sells a refrigerator for

0.3

(a) 5

(d)

each number in scientific notation:

(69)

(a)

18. 1751

19.

Percent (b)

(c)

12

17. Write

sale?

Decimal

Fraction

(48)

of the price of

30 x 10

f

(b)

30 x 10

Find the area of the trapezoid shown. Dimensions are in meters.

Each edge of a cube measures

-6

pr

5 inches.

(67, 70)

(a)

What

is

the

(b)

What

is

the surface area of the cube?

volume

of the cube?

20. In a bag are 100 marbles: 10 red, 20 white, 30 blue, (36)

and 40

one marble is drawn from the bag, what is the probability that the marble will not be red, white, or blue?

green.

If

For problems 21-23, solve and check each equation. Show each step. 21. (Inv. 7)

17a = 408

22. (Inv. 7)

|m 8

= 48

545

Lesson 80

23. 1.4 =

X - 0.41

Simplify: 2

3

+ 4

•

5

-

2

•

3"

25.

24. -.25

28. 2*

(43)

6 oz

n 2/.

-

26. (56)

-.4

•

10 lb

-

rl

x 1.4

7

,

1

cm 2

10

mm

10

mm

•

•

1

cm

1

cm

7 lb 11 oz

7l 2

3 {

30. Triangle

29.

|9)

ABC

with vertices

C(2.0) is reflected in the image lA'B'C.

2

at

.4(0.2).

x-axis.

5(2.2).

Draw A ABC and

and its

546

Saxon Math 8/7

Focus on

Using a Compass and Straightedge, Part 2 we used

In Investigation 2

a

compass

to

draw

circles,

and we

used a compass and straightedge to inscribe a regular hexagon and a regular triangle in a circle. In this investigation we will use a compass and straightedge to bisect (divide in half) a line segment and an angle. We will also inscribe a square and a regular octagon in a circle. Materials needed:

Bisecting a line

segment

Use

•

Compass

•

Ruler or straightedge

•

Protractor

a metric ruler to

endpoints

A

and

draw

a

segment 6

cm

long. Label the

C.

•

•

A

C

Next open a compass so that the distance between the pivot point and pencil point is more than half the length of the segment to be bisected (in this case, more than 3 cm). You will swing arcs from both endpoints of the segment, so do not change the compass radius once you have it set. Place the pivot point of the compass on one endpoint of the segment, and make a curve by swinging an arc on both sides of the segment as shown.

Investigation 8

Then move

547

the pivot point of the compass to the other

endpoint of the segment, and, without resetting the compass, swing an arc that intersects the other arc on both sides of the segment. Draw a line through the two points where the arcs intersect to divide the original segment into two parts. Label the point where the line intersects the segment point B.

A

1.

Use

2.

Where

a metric ruler to find

formed.

AB and BC.

the line and segment intersect, four angles are

What

is

the measure of each angle?

geometric figures just constructed the perpendicular

Using a compass and straightedge is

called construction.

You

to create

bisector of a segment.

3.

Why

is

the line you constructed called the perpendicular

bisector of the segment?

Inscribing a square in a circle

We

can use a perpendicular bisector to help us inscribe a square in a circle. Draw a dot on your paper to be the center of a circle. Set the distance between the points of your

compass

to 2

cm. Then place the pivot point of the compass

548

Saxon Math 8/7

on the dot and draw a diameter of the

Use

circle.

a straightedge to

draw

a

circle.

The two points where the diameter

intersects the circle are

Open

the compass a little more than the radius of the circle, and construct the perpendicular bisector of the diameter you drew. the endpoints of the diameter.

/ •

V\ Make

/ •




j

2

2

36

•

ft

ft

2

LESSON PRACTICE Practice set*

a.

Using 3.14

for n, calculate to the nearest square foot the

area of circle

(c)

in this lesson's example.

Find the area of each b.

circle

"X

C. / {

8

cm

\

•

J

Use 3.14 \wn.

——

/4 cm

Leave n as

d. \

8

/

cm

\

• J

n.

Use

—

22

for n.

MIXED PRACTICE Problem set

1. l70)

2. 28)

Find the volume of this rectangular prism. Dimensions are in feet.

2.5

The heights of the five basketball starters were 6'3", 6'5", 5'11", 6'2", and 6'1". Find the average height of the five starters. (Hint: Change all measures to inches before dividing.)

5 63

Lesson 82

3. 541

4. 50)

5.

problem. The student-teacher ratio at the high school was 20 to 1. If there were 48 high school teachers, how many students were there?

Use

a ratio

box

to solve this

An

inch equals 2.54 centimeters. Use a unit multiplier to convert 2.54 centimeters to meters.

Graph each inequality on

a separate

number

line:

(78)

(a)

6. 72)

Use

x < -2 a case 1-case 2 ratio

box

x >

to solve this

problem. Don's

heart beats 225 times in 3 minutes. At that rate,

many 7.

(b)

how

times will his heart beat in 5 minutes?

Diagram

statement.

this

Then answer

the questions

that follow.

Two fifths

of the students in the class were boys. There were 15 girls in the class.

8.

(a)

How many

fb)

What was

students were in the class?

the ratio of girls to boys in the class?

Compare: x 2 - y 2

(79)

9. (Inv. 1)

What

percent

O

of

i

x +

this

y)i x

-

circle

y)

is

shaded?

Ob

- b

10.

Compare: a

11.

Find the circumference of each

if

a

is

negative

circle:

(66)

Use 3.14

for n.

Use

—

for n.

564

Saxon Math 8/7

12.

Find the area of each

problem

circle in

11.

(82)

13.

Complete the

table.

(48)

Fraction

Decimal 1.6

(a)

(b)

(d)

(c)

14.

Percent

1.6%

Write an equation to solve this problem:

(60)

How much money 15. Write

is

6.4% of $25?

each number in scientific notation:

(69)

(a)

Use

12 x 10 5

ratio

(b)

5 12 x 10"

boxes to solve problems 16 and 17.

16. Sixty-four percent of the students correctly described the t81)

process of photosynthesis. If 63 students did not correctly describe the process of photosynthesis, how many students did correctly describe the process?

40 percent of her book to read. If she has read 180 pages, how many pages does she still have

17. Ginger still has

m

to read?

18. (75)

Find the area of the figure shown. Dimensions are in inches. Corners that look square are square.

12

19.

m

The coordinates of the vertices of AXYZaie X(4, 3), 7(4, 1), and Z(l, 1). Draw AXYZ and its image AX'Y'Z' translated 5 units to the left and 3 units down. What are the coordinates of the vertices of AXYZ'?

prime factorization of the two terms of Then reduce the fraction.

20. Write the ,24)

fraction.

240 816

this

Lesson 82

21. dnv. 2)

The

figure

regular

illustrates

hgxaggn ABCDEF inscribed circle with center at point M. (a)

How many

illustrated

in

a

chords

are diameters? (b)

How many

illustrated

chords

are not diameters? (c)

What

is

the measure of central angle AXIB?

(d)

What

is

the measure of inscribed angle

22. Write

ABC?

100 million in scientific notation.

(51)

For problems 23 and 24, solve and check the equation. SI each step. 23. (Inv. 7)

-x

= 36

24. 3.2 + a = 3.46

4

(Inv. 7)

Simplify:

\3~ +

4'

25.

26. (8

(52)

27. (43, 45)

-

2

3)

(52)

3^

-r

(7

-f

(decimal answer)

0.2)

2

28. 4.5

+ 2— —

3

(mixed-number answ er) r

3

(43)

29. (a)

tffcfl {—ZJ

(73)

(b)

(-2)(+3)(-4)

30. (a) (-3)

+ (-4) - (-2)

(68)

(b)

(-20) + (+30)

-

|-40|

-

(3

-

:

8)

566

Saxon Math 8/7

LESSON

83

Multiplying Powers of 10 Multiplying Numbers in Scientific Notation

4

•

WARM-UP Facts Practice: Area (Test R)

Mental Math: (+3)

b.

6 6.75 x 10

$120

e.

500

a.

(-60)

d.

15%

g.

At 60 mph, how

4-

of

mg tog

_ m

100 150

c.

24

f.

30

is

| of n.

l\ hours?

far will a car travel in

Problem Solving:

How many

different triangles of

any

size are in this

figure?

NEW CONCEPTS Multiplying powers of 10

From our

earlier

work with powers 1q3

means 10

we remember that

of 10,

.

10

.

10

10

•

10

•

and 10 4 means 10

Now let us

10

multiply these two powers of 10. 10 3 10

We

•

see that 10

write this as 10

•

3 •

10

10 4

•

•

10

•

10

•

10

•

10

•

10

10 4 means 7 tens are multiplied.

We

can

7 .

j

10 3

10 4 = 10 7

•

j

As we focus our

attention

on the exponents, we see

3

The above example

When we

that

+ 4 = 7

illustrates

an important rule of mathematics.

multiply powers of 10,

we add

the exponents.

567

Lesson 83

Multiplying

numbers

in

scientific

notation

To multiply numbers written in scientific notation, we multiply the decimal numbers to find the decimal-number part of the product. Then we multiply the powers of 10 to find the power-of-10 part of the product. We remember that

when we multiply powers Example

1

Solution

we add

of 10,

the exponents.

7 Multiply: (1.2 x 10 5 )(3 x 10 )

We

multiply 1.2 by 3 and get 3.6. Then 7 10 and get 10 12 The product is

we

multiply 10 5 by

.

12 3.6 x 10

Example 2 Solution

Multiply:

(4

6

x 10 )(3 x 10

5 )

We

multiply 4 by 3 and get 12. Then and get 10 11 The product is

we

multiply 10 6 by 10 5

.

12 x 10 11

We rewrite this

expression in proper scientific notation.

Example

3

Solution

Multiply:

(2

1

x 10

(1.2

x 10

)

11

= 1.2 x 10 12

7 x 10" 5 )(3 x 10" )

We

7 5 multiply 2 by 3 and get 6. To multiply 10" by 10~ add the exponents and get 10~ 12 Thus the product is

,

we

.

12 6 x 1(T

Example 4 Solution

Multiply:

(5

10" 8 )

3

x 10 )(7 x

We

multiply 5 by 7 and get 35. 5 10" get The product is

We

8 multiply 10 3 by 10" and

.

5 35 x 10"

We rewrite this

expression in scientific notation.

(3.5

x 10

1

x 10

)

-5

= 3.5 x 10" 4

ESSON PRACTICE Practice set*

Multiply. Write each product in scientific notation. 6 x 10 )(1.4 x 10

3

a.

(4.2

b.

(5

x

c.

(4

7 x 10" 3 )(2.1 x 10~ )

d. (6 x

7 10 5 )(3 x 10

)

)

10" 2 )(7 x 10" 5 )

568

Saxon Math 8/7

MIXED PRACTICE Problem set

1. 1461

The 16-ounce box costs $1.12. The 24-ounce box costs $1.32. The smaller box costs how much more per ounce than the larger box?

2. l65)

box to solve this problem. The ratio of good apples to bad apples in the basket was 5 to 2. If there were 70 apples in the basket, how many of them were good?

Use

a ratio

average score after fifteen tests was 82. Her average score on the next five tests was 90. What was her average score for all twenty tests?

3. Jan's (55)

4. (53}

5.

earns $6 per hour at a part-time job. How does he earn if he works for 2 hours 30 minutes?

Hakim

Use

a unit multiplier to convert 24 shillings to pence

shilling = 12 pence).

(1

6.

much

Graph x < -1 on

a

Use a case 1-case

box to solve what number?

number

line.

(78)

7. (72)

8.

20

2 ratio

is

to 12 as

If

a = 1.5, what does 4a + 5 equal?

is

to

this

problem. Five

(41)

9. l22)

Four fifths of the football team's 30 points were scored on pass plays. How many points did the team score on pass plays?

10.

Compare: x(x +

y]

Qx

2

+ xy

if

x and y are

integers

(79)

11.

Find the circumference of each

circle:

(66)

(a)

Leave k as

n.

12.

Find the area of each

(82)

indicated values for n.

Use

circle in

—

22

for n.

problem 11 by using

the

569

Lesson 83

13.

The edges

cube are 10

of a

cm

long.

(67, 70)

14.

(a)

What

is

the

(b)

What

is

the surface area of the cube?

Complete the

volume

table.

of the cube?

12

(46J

What

is

250% (d)

(c)

the sales tax on an S8.50 purchase

ratio

16.

Monifa found

boxes

to solve

problems 16 and

the sales-tax

17.

minutes of commercials aired during every hour of prime-time programming. Commercials were shown for what percent of each hour? that

12

17. Thirty percent of the boats that traveled {81)

if

rate is 6i°o?

Use {81

Percent

(b)

(a)

15.

Decimal

Fraction

(48)

Monday were steam-powered.

up the

river

on

42 of the boats that traveled up the river were not steam-powered, how many If

boats were there in all?

prime factorization of the numerator and denominator of this fraction. Then reduce the fraction.

18. Write (24]

the

420 630

19.

Find the area of the trapezoid

40

at

right.

24

m

25 35

and ZB of \ABC are congruent. The measure of ZE : Find the measure of is 54

20. In this figure. 40

ZA

.

(a)

ZECD.

(b)

ZECB.

(c)

ZACB.

(d)

ZBAC.

m

m

m

570

Saxon Math 8/7 21. Describe the rule of this function, l58)

In

Out

and find the missing numbers. 21 5

11

7

15

2

5

-5 22. Multiply. Write (83)

(a)

(3

each product in scientific notation.

4 x 10 )(6 x 10

5 )

(b)

(1.2

x 10

_3

)(4

x 10

-6 )

For problems 23 and 24, solve and check the equation. Show each step. 23.

?

b -

1=-

3

(Inv. 7)

1

24. 0.4y = 1.44

= 4^ 2

(Inv. 7)

Simplify: 25. 2

3

+ 22 + 21

4-

2°

+ 2" 1

(52, 57)

26. 0.6 x 3^ 3 (43)

28. 7T7

24

(b)

-r

2

_7_

60

(-3)(-4)(-5)

29. (a) (-3) + (-4)

-

(-5)

(68)

(b)

30.

m

(-15)

-

(+14) + (+10)

The coordinates of the vertices ofAPQR are P (0, 1), Q (0, 0), and R (-2, 0). Draw the triangle and its image kP'Q'R' after a 180° clockwise rotation about the origin.

coordinates of the vertices of AP'Q'R'?

What

are the

571

Lesson 84

E S S

O N

84

Algebraic Terms

ARM-UP Facts Practice: Area (Test R)

Mental Math:

-

(-12)

b.

25 2

cm

e.

1.5

a.

(-12)

d.

3.14 x 30

g.

12 x 12,

-

10,

4,

+

c.

cm

1,

to

mm

x 2, + 3,

f.

r

3,

6m 30 x

is

5,

10 = 32 | of

-

m t

6,

f"

Tabletop

R

1,

Problem Solving:

A

rectangular tablecloth was draped over a rectangular table. Eight inches of cloth hung over the left edge of the table, 3 inches over the back, 4 inches over the right edge, and 7 inches over the front. In which directions (L, B, R, and F) and by how many inches should the tablecloth be shifted so that equal amounts of cloth hang over opposite edges of the table?

B

;w CONCEPT

We

have used the word term in arithmetic to refer to the numerator or denominator of a fraction. For example, we reduce a fraction to its lowest terms. In algebra term refers to a part of an algebraic expression or equation. An algebraic expression

may

contain one, two, three, or more terms. Some

Type

of

Expression

Algebraic Expressions

Number

of

Terms

Example

monomial

1

-2x

binomial

2

2 a 2 - 4b

trinomial

3

3x 2 - x - 4

from one another in an expression by plus or minus signs that are not within symbols of inclusion.

Terms

are separated

572

Saxon Math 8/7

Here we have separated the terms of the binomial and trinomial examples with slashes: a

2

/

3x 2

- 4b 2

/

- x

/

- 4

Each term contains a signed number and may contain one oi more variables (letters). Sometimes the signed-number part is understood and not written. For instance, the understood signed-number part of a 2 is +1 since a 2 = +la 2 Likewise, the term -x is understood to mean -lx. When a term is written without a number, it is understood that the number is 1. When a term is written without a sign, it is understood that .

not necessary for a term to contain a variable. The third term of the trinomial above is -4. A term that does not contain a variable is often called a constant term, because its value never changes. the sign

is

positive.

It is

Constant terms can be combined by algebraic addition.

3x+3-l

=

3x+2

added +3 and -1

Variable terms can also be combined by algebraic addition if they are like terms. Like terms have identical variable parts. That is, the same variables with the same exponents appear in the terms. The terms -3xy and +xy are both xy terms. They are like terms and can be combined by algebraically adding the signed-number part of the terms.

-3xy + xy = -2xy

The signed number -3xy and -i-lxy. Example

1

part of

+xy is

+1.

We get -2xy by

adding

Collect like terms in this algebraic expression.

3x + y + x - y Solution

There are four terms in this expression. There are two x terms and two y terms. We can use the commutative property to rearrange the terms.

3x + x + y - y

Adding +3x and +lx we we get Oy, which is 0.

get +4x.

Then adding +ly and -ly

3x + x + y - y

4x + 4x

Lesson 84

Example

2

Collect like terms in this algebraic expression:

3x + 2x 2 + 4 + x 2 - x Solution

5 73

1

x 2 terms. x terms, and constant terms. Using the commutative property we arrange them to put like terms next to each other. In this expression there are three kinds of terms:

2x 2 + x 2 + 3x - x + 4 -

Now we

1

collect like terms.

2x 2 + x 2 + 3x - x + 4 3x 2 + 2x +

1

3

Notice that x 2 terms and x terms are not like terms and cannot be combined by addition. There are other possible arrangements of the collected terms, such as the following:

2x + 3x 2 + Customarily, however,

we

3

arrange terms in descending order

of exponents so that the term with the largest exponent

and the constant term

on the right. written without a constant term is understood the

left

is

is

on

An to

expression have zero as

a constant term.

ISSON PRACTICE Practice set

Describe each of these expressions as a monomial, a binomial, or a trinomial: a.

x 2 - y2

b.

3x 2 - 2x -

c.

-2x 3 yz2

d.

-2x 2 y - 4xy 2

1

Collect like terms: e.

2 3a + 2a 2 - a + a

g.

3

f.

2 + x 2 + x - 5 + 2x

h.

5xy - x + xy - 2x 3k +

1.4

-K +

2.8

IIXED PRACTICE

Problem set

1.

m

An

: increase in temperature of 10 on the Celsius scale corresponds to an increase of how many degrees on the

Fahrenheit scale? 2. (84)

Collect like terms:

2xy + xy - 3x + x

574

Saxon Math 8/7

3.

Refer to the graph below to answer (a)-(c).

(Inv. 4)

Daily High

Temperature

80

Mon.

(a)

(b)

Wed.

Tues.

What was the range of from Monday to Friday?

Which day had

Thu.

Fri.

the daily high temperatures

the greatest increase in temperature

from the previous day? (c)

Wednesday's high temperature was how much lower than the average high temperature for these 5 days?

4.

Frank's scores on ten tests were as follows:

(Inv. 4)

90, 90, 100, 95, 95, 85, 100, 100, 80,

For this (c)

5. (65)

mode, and

Use

a ratio

rowboats

number

6.

(a)

mean,

(b)

median,

(d) range.

box

solve this problem.

to

to sailboats in the

bay was

The

sailboats

ratio

of

3 to 7. If the total

and sailboats in the bay was were in the bay?

of rowboats

how many

dnv. 3)

find the

set of scores,

100

210,

Recall that the four quadrants of a coordinate plane are num bered i s t 2nd, 3rd, and 4th in a counterclockwise direction beginning with the upper right quadrant. In which quadrant are the x-coordinates negative and the y-coordinates positive? ?

7. {72]

8. 1711

Write a proportion to solve this problem. how much would 10 cost? Five

eighths

of

the

If

members supported

whereas 36 opposed the supported the treaty?

treaty.

4 cost $1.40,

the

treaty,

How many members

575

Lesson 84

9.

Evaluate each expression for x =

o:

52

(a)

10.

x 2 - 2x +

Compare:

fQg

8

if

79)

11. (a) (66' 82)

(b)

-

(b) (x

1

=

l)

2

1

Find the circumference of the circle shown. Find the area of the

circle.

Use 3.14 for x.

12.

Use a unit multiplier

13.

Draw manv

m

to convert 4.8

a rectangular prism.

A

meters to centimeters.

rectangular prism has

faces?

Fraction

DECIMAL

li5

m

Percent [b]

l.S

[d]

(c]

15.

how

:

:

Write an equation to solve this problem. A merchant priced a product so that 30°o of the selling price is profit. If the product sells for S18.00. how much is the merchant's profit?

12 1

16. Simplify:

boxes to solve problems 17 and

Use

ratio

17.

When

"

:

m was

18.

open. 36 pigeons flew the coop. If this was 40 percent of all pigeons, how many pigeons were the door

left

originally in the coop?

18. Sixtv percent of the saplings r:

there

were 300 saplings in

3 feet tall?

all.

were

3 feet tall or less. If

how many were more

than

Saxon Math 8/7

19. 119,75)

A

square sheet of paper with a p er i me t er f 43 irL nas a corner cut off, forming a pentagon as shown.

What

(a)

the perimeter of the

is

pentagon?

What

(b)

20.

The

the area of the pentagon?

is

been divided into sevei the central angles of which have the following

face of this spinner has

sectors,

measures:

A

60

E

75'

B 90

(

F

40

c

c

C

45'

G

20

D

c

the spinner is spun once, will stop in sector If

A?

(a)

(b)

30°

what

is

the probability that

C?

(c)

Then

21. Describe the rule of this sequence. [2]

three

il

E? find the next

numbers of the sequence. 1, 3, 7,

15, 31,

...

22. Multiply. Write each product in scientific notation. (83)

x 10~ 3 )(3 x 10 b )

(1.5

(a)

(b)

(3

x 10 4 )(5 x 10

5 )

Find each missing exponent:

^ 2

23.

10

(a)

2

10

•

2 •

10

2

= 1(P

(b)

10

(57, 83)

6

=

lcP I

For problems 24 and 25, solve and check the equation. Show each step. 24. b

- 4.75 =

5.2

25.

(Inv. 7)

(Inv. 7)

= 36 %y O

Simplify: 26.

V5

Z

-4^+2'

27. 1

(52)

28. (43)

29

.

— 10 9

(a)

-r

1 2—

- 45

mm

24 (decimal answer)

•

4

t«

(b)

(-3) (+4)

(73)

30.

m

(32)

(a)

(+30)

(b)

(-3)

-

(-50)

-

(68)

-

(-4)

-

(5)

(+20)

(+3)(-5)(+2)

577

Lesson 85

E S S

O N

85

Order of Operations with Signed Numbers

•

Functions, Part 2 'ARM-UP

Facts Practice: +

-x

-r

Integers (Test P)

Mental Math: a.

r v«

(+12)(-6)

b. (4

1

d.

_ -

1.5

80 n

km to m

10

x

$12

is

What

3

)(2

| of

x 10

6 )

how much money

\ of

$12?

e.

0.8

g.

Find the perimeter and area of a square with sides 2.5

f.

is

(m)?

m long.

Problem Solving: There are three numbers whose sum is 180. The second number is twice the first number, and the third number is three times the first number [n, In, and 3n). Find the three numbers. (Try guess and check; then try writing an

+ 180

equation.)

EW CONCEPTS Order of operations with signed

numbers

Example

1

To simplify expressions

that involve several operations,

perform the operations in a prescribed order. We have practiced simplifying expressions with whole numbers. In this lesson we will begin simplifying expressions that contain both whole numbers and negative numbers.

Simplify: (-2) + (-2)(-2)

- (=3 (+2)

Solution

we

First

we

multiply and divide in order from

(-2)

(-2) + (-2H-2) (+2)

(-2)

+

(+4)

-

(-1)

left to right.

Saxon Math 8/7

Then we add and

subtract in order from

+ (+4) -

(-2)

left to right.

(-1) ;

-

(+2)

(-1) I

+3 Mentally separating an expression into expression easier to simplify. (-2)

First

we

/

+ (-2H-2)

simplify each term; then

terms can make

its

/-

{=fl

we combine

(-2)

/

+ (-2H-2)

-2

/

+ 4

ai

the terms.

/ - tfl +

/

1

+3

Example 2

Simplify each term. Then combine the terms. -3(2

Solution

-

4)

-

4(-2)(-3)

+

(

~ 3)( ~ 4)

We

separate the individual terms with slashes. The slashes precede plus and minus signs that are not enclosed by

parentheses or other symbols of inclusion. -3(2

Next

we

-

4)

4(-2}(-3)

J

+

M) M) 2

simplify each term.

-3(2 - 4)

Now we

J

-

/

-

4(-2)(-3)

/

+

(

" 3) ~ 4) (

2

^

-3(-2)

/

+ 8(-3)

/

+

+6

/

- 24

/

+6

combine the simplified terms.

+6-24+6 -18 + 6 -12

Lesson 85

Example 3 Solution

Simplify: (-2)

-

[(-3)

-

5 79

(-4)(-5)]

There are only two terms, -2 and the bracketed quantity. By the order of operations, we simplify within brackets first, multiplying and dividing before adding and subtracting. (-2) /

-

-

(-2) /

- (-4X-5)]

[(-3)

-

[(-3)

-

(-2) /

(+20)]

(-23)

+23

(-2) /

+21 Functions, part 2

We

remember

between two sets of numbers. We have practiced finding missing numbers in functions when some number pairs have been given. For instance, the missing numbers in the functions on the left and the right below are 14 and 7, respectively. that a function

3

4 7

Out

F

In

— — — —

14

C I

O

N

— UN — — T — — — NO

7

C

6

10

8

—

Out

F

In

U N

— — T —

a relationship

is

5

I

3 2

We found the missing numbers by first finding the rule

of the

The rule of the function on the left is multiply the "in" number by 2 to find the "out" number. The rule of the function on the right is subtract 7 from the "in" number to

function.

find the "out" number.

Often the rule of a function is expressed as an equation with x standing for the "in" number and y standing for the "out" number. The equation for the rule of the function on the left is

y = 2x The

rule of the function

on the

right is

y = x —

7

we will tables when

Beginning with this lesson

practice finding missing

numbers

the rule

equation.

in function

is

given as an

580

Saxon Math 8/7

Example 4

Find the missing numbers in the table using the function

y = 2x + X

rule.

1

y

4 7

Solution

stands for the "out" number. The letter x stands for the "in" number. We are given three "in" numbers and are asked to find the "out" number for each by using the rule of the function. The expression 2x + 1 shows us what to do to find y, the "out" number. It shows us we should multiply the

The

letter

y

x number by multiply by 2

and then add and add 1.

2

y = 2x + y =

7=8 y=9

+

The

first

1

added

+

2(7)

third

x number

1

added

15 is 0.

2(0)

7=0 7=1 The missing numbers

substituted

multiplied

1

We multiply by

7 = 2x + 7 =

The next x number

1

7 = 14 +

The

We

multiplied

1

7 = 2x +

7 =

is 4.

substituted

We find that the y number is 9 when x is 4. is 7. We multiply by 2 and add 1.

7 =

x number

1

+

2(4)

1.

+

2

and add

1

+

1

1

substituted

multiplied

added are 9, 15,

and

1.

1.

581

Lesson 85

ESSON PRACTICE Practice set

Simplify: a.

(-3)

+

(-3) (-3)

-

(-3)

- (-5H-6)]

(-3)

-

[(-4)

d. (-5)

-

(-5) (-5)

b.

(+3) c.

- (-4H-5)]

(-2)[(-3)

+ I-5

Find the missing numbers in each table by using the function rule:

= 3x -

X

i

y = X y

f.

y

6

3

1

1

g.

y = X

y

7

1

1

n

1

1

4

h. Jacinta

- x

8

4

i

i

when x was and when x was 3, y

studied a function and found that

y was 1; when x was 2, y was 4; was 9. Make a table of x, y pairs for 1,

this function,

and

above the table write an equation that expresses the function rule.

IXED PRACTICE

Problem set

1. dnv.4)

Find the

(a)

mean,

(b)

median,

(c)

mode, and

2. (65)

3. (54)

range of

the following scores: 70, 80, 90, 80, 70, 90, 75, 95, 100,

Use

(d)

ratio

90

boxes to solve problems 2-4:

The team's ratio of games won to games played was 3 to 4 jf t ^ e team pi a y e d 24 games, how many games did the team fail to win?

Mary was chagrined

dandelions to If there were 44 dandelions were

to find that the ratio of

marigolds in the garden was 11 to 4. marigolds in the garden, how many there?

sound travels 2 miles sound travel in 1 minute?

4. If 1721

5.

Use a unit multiplier

in 10 seconds,

6.

Graph x >

on

a

far

does

to convert 0.98 liter to milliliters.

(50)

(78)

how

number

line.

Saxon Math 8/7

7. (71)

Diagram

Then answer

statement.

this

the question

that follow.

thousand dollars was raised in the charity drive. This was seven tenths of the goal Thirty-five

8.

(a)

The goal money?

(b)

The drive

of the charity drive

fell

short of the goal by

Oa

Compare: 2a

was

2

if

a

is

how mud

to raise

what percent?

whole number

a

(79)

9.

The radius

of a circle

is

4 meters. Use 3.14 for n to find the

(66, 82)

10. (Inv

l >

11.

2.5

g e.

g.

1.87

m to cm

10%

of 80, x

(2.5

d.

Estimate

+

1,

y[

X

+

7,

7§%

of $8.29.

Sl.OOislofm. o

f.

3,

4 H x 10~ )(3 x 10 )

b.

1,

-r

2,

-r

2,

\

,

x 10, +

2,

- 4

Problem Solving: Here are the

front, top,

and side views of an

object.

Construct this

object using 1-inch cubes, or sketch a three-dimensional view.

Then

calculate the object's volume. 2

in.

1

1

2

in.

in.

2 1

in.

in.

1

in.

2

Front

1

in.

1

in.

in.

in.

Top

Right Side

JEW CONCEPTS Diagonals

Recall that a diagonal of a polygon

is

a line segment that passes

through the polygon between two nonadjacent vertices. In the figure below, segment ^4Cis a diagonal of quadrilateral ABCD.

Example

1

From one vertex of ABCDEF, how many

regular

diagonals can be

drawn? (Trace the hexagon and your answer.)

hexagon illustrate

B

Saxon Math 8/7

Solution

Interior

angles

We We

can select any vertex from which to draw the diagonals. choose vertex A. Segments AB and AF are sides of the hexagon and are not diagonals. Segments drawn from A to C, D, and E are diagonals. So 3 diagonals can be drawn. F

A

D

C

Notice in example 1 that the three diagonals from vertex divide the hexagon into four triangles. We will draw arcs emphasize each angle of the four triangles.

Angles that open interior angles.

We

to

F

A

D

C

A to

the interior of a polygon are called

see that

ZB

of the

ZB of ZBCA of AABC and

hexagon

AABC. Angle C of the hexagon includes ZACD of AACD. Are there any angles of the

is

also

four triangles that

hexagon?

are not included in the angles of the

we may not know the measure of each angle of each we nevertheless can conclude that the measures of the

Although triangle,

six angles of a

hexagon have the same

the angles of four triangles, F

which

measures

of

4 x 180° = 720°.

is

A The sum

B

interior

of the

measures

of the six

angles of a hexagon

4

D

total as the

x

is

720°.

180° = 720°

C

hexagon ABCDEF is a regular hexagon, we can calculate the measure of each angle of the hexagon. Since

Example 2

Maura inscribed

a regular

hexagon in a

Find the measure of each angle of the regular hexagon ABCDEF.

circle.

F

d

.rx

6 03

Lesson 89

Solution

From

the explanation above we know that the hexagon can be divided into four triangles. So the sum of the measures of the angles of the hexagon is 4 x 180°, which is 720°. Since the hexagon is regular, the six angles equally share the available 720°. So we divide 720° by 6 to find the measure of each angle.

720°

We Example

3

-r

6 = 120°

find that each angle of the hexagon measures 120°.

Draw sum

and one of measures of the

a quadrilateral

of

the

its

diagonals.

interior

What

angles

is

of

the the

quadrilateral?

Solution

We

draw

a four-sided

polygon and a diagonal.

Although we do not know the measure of each angle, we can find the sum of their measures. The sum of the measures of the angles of a triangle is 180°. From the drawing above, we see that the total measure of the angles of the quadrilateral equals the total measure of the angles of two triangles. So the sum of the measures of the interior angles of the quadrilateral is 2

Exterior

angles

x 180°

= 360°

example 2 we found that each interior angle of a regular hexagon measures 120°. By performing the following activity, we can get another perspective on the angles of a polygon. In

Activity: Exterior Angles

Materials needed: •

A length of string

•

Chalk

•

Masking tape

(5 feet

or more)

(optional)

Before performing this activity, lay out a regular hexagon in the classroom or on the playground. This can be done by inscribing a hexagon inside a circle as described in

Saxon Math 8/7

Use the string and chalk to sweep out the circle and to mark the vertices and sides of the hexagon. If desired, mark the sides of the hexagon with masking tape. Investigation

2.

After the hexagon has been prepared, walk the perimeter the hexagon while making these observations: 1.

2.

of

you were facing when you started around the hexagon as well as when you finished going around the hexagon after six turns. How much you turned at each "corner" of the hexagon. Did you turn more than, less than, or the same as you would have turned at the corner of a square?

The

direction

Each student should have the opportunity

to

walk

the

perimeter of the hexagon.

Going around the hexagon, did not turn,

we

we

turned would continue going Path

if

we

at

every corner.

If

we

straight.

did not turn

The amount we turned at the corner hexagon equals the measure of the hexagon at that vertex.

in order to stay

on

the

exterior angle of the

We

can calculate the measure of each exterior angle of a regular hexagon by remembering how many turns were required in order to face the same direction as when we started. We remember that we made six small turns. In other words, after six turns we had completed one full turn of 360°. If all the turns are in the

exterior angles of any

same

direction, the polygon is 360°.

sum

of the

Lesson 89

Example 4

Solution

What

the measure of each exterior angle of a regular hexagon?

605

is

/\

Traveling all the way around the hexagon completes one full turn of 360°. Each exterior angle of a regular hexagon has the same measure, so we can find the measure by dividing 360° by 6. 360°

We

find

measures

that 60°.

-r

6 = 60°

each exterior angle of a regular hexagon

Notice that an interior angle of a polygon and its exterior angle are supplementary, so their combined measures total 180°.

ESSON PRACTICE Practice set

a.

Work examples 1-4 from

this lesson if

you have not

already.

b.

c.

Trace this regular pentagon. How many diagonals can be drawn from one vertex? Show your work.

The diagonals drawn into

d.

how many

What

is

the

in

problem b divide the pentagon

triangles?

sum

of the measures of the five interior

angles of a pentagon?

e.

What

is

the measure of each interior angle of a regular

pentagon?

606

Saxon Math 8/7

f.

What

g.

What

each exterior angle of a regular pentagon? is

measure

the

of

sum

of the measures of an interior and exterior angle of a regular pentagon?

the

is

MIXED PRACTICE Problem set

1. (72)

2. l54)

3. 1701

ratio

4.

5.

m

6.

to solve this

were lions, tigers, and bears. The ratio of lions to tigers was 3 to 2. The ratio of tigers to bears was 3 to 4. If there were 18 lions, how many bears were there? Use a ratio box to find how many tigers there were. Then use another ratio box to find the number of bears. In the forest there

Kwame measured the shoe box and found that it was 30 cm long, 15 cm wide, and 12 cm tall. What was the volume

l44)

box

problem. Jason's remotecontrol car traveled 440 feet in 10 seconds. At that rate, how long would it take the car to travel a mile?

Use a

A

of the shoe box?

baseball player's batting average

is

a ratio found by

dividing the number of hits by the number of at-bats and writing the result as a decimal number rounded to the nearest thousandth. If Erika had 24 hits in 61 at-bats what was her batting average?

Use two unit multipliers

to convert 18 square feet

tc

square yards.

On

a

number

line,

graph the integers greater than -4.

(86)

7. {71]

Diagram

this

statement.

Then answer

the question!

that follow.

Jimmy bo ugh tthe shirt for $12.

This was

| of the

regular price. (a)

What was

(b)

Jimmy bought price?

the regular price of the shirt?

the shirt for

what percent of the regula

Lesson 89

8.

Use the

figure

below

to find the

measure of each

607

angle.

(40)

9.

(a)

Za

(a)

What

l66 823 -

(b)

Zb

(c)

Zc

(d)

Zd

the circumference of

is

this circle? (b)

What

is

the area of this circle?

—

22

Use

for n.

9110. Simplify:

11. Evaluate: (52)

12.

°^ + Q L a + b

Compare: a 2

(79)

13.

Oa

Complete the

table.

(48)

a = 10

if

if

Fraction 7

Decimal

Percent (b)

(a)

875%

(d)

(c)

(17)

5

a = 0.5

8

14.

and b =

At three o'clock and at nine o'clock, the hands of a clock form angles equal to \ of a circle. (a)

At which two hours do the hands of a clock form angles equal to | of a circle?

(b)

The angle described

in part

(a)

measures

how many

degrees?

Use

ratio

boxes to solve problems 15 and

3000 fast-food customers ordered many of the customers ordered a

15. Forty-five percent of the (81)

a hamburger.

hamburger?

How

16.

608

Saxon Math 8/7

16. (81)

The

sale price of

sale price

75%

$24 was

of the regular price. The dollars less than the regular

was how many

price?

17. Write

an equation

to solve this

problem:

(77)

Twenty 18. (a) Trace this

and draw

is

what percent of 200? 30

isosceles trapezoid

its

line of

symmetry.

24

i

26\

(b)

Find the area of the trapezoid.

mm

*

|

1891

20. 1851

What

is

measure

the

/ 26

mm

10

19.

mm

U '

mm

mm

each

of

exterior angle of a regular triangle?

Find the missing numbers in the table by using the function rule.

y =

ix

X

7

12

1

1

9

1

1

u 21. Multiply. Write the

6

product in scientific notation.

(83)

(1.25 x 10

22.

m

The lengths and 10 cm. (a)

Draw

(b)

Can

of

_3

)(8

two sides of an

the triangle and find

there be

5 x 10" )

isosceles triangle are 4

its

perimeter.

more than one answer?

Why or why not?

For problems 23 and 24, solve and check the equation. each step. 23. (Inv. 7)

-p 9

= 72

24. 12.3 = 4.56 + (Inv. 7)

cm

/

Show

Lesson 89

25. Collect like terms:

2x + 3y - 4 + x - 3y -

1

(84)

Simplify: 26>

9-8-7-6 6

28.

.

5

•

13- -

3 2 x 4

(a)

(as)

(b)

-

4.75 +

^

x 1q2

(mixed-number answer)

+ t-4)(-6)

(-3)

+ (-4) - (-6)

(-5)

-

(+6)(-2) + (-2)(-3)(-l)

2

30. (a) (3x )(2x) (87)

(b)

_2

f43, 57J

3

(43)

29

2?

(~2ab){-3b 2 )(-a)

609

.

610

Saxon Math 8/7

LESSON

90

Mixed-Number

Coefficients Negative Coefficients

WARM-UP Facts Practice: Percent-Decimal-Fraction Equivalents (Test Q)

Mental Math: a.

(-50)

-

(-30)

b.

(4.2

c.

4w -

8 = 36

d.

Estimate

x 10

-6 )(2

15%

4 x 10" )

of $23.89.

f. $1.00 is | of m. 800 g to kg A cube with edges 10 inches long has a volume of

e.

g.

how many

cubic inches?

Problem Solving: If

four people shake hands with one another,

number

we can

by drawing four dots (for people) and connecting the dots with segments (for handshakes). Then we count the segments (six). Use this method to count the number of handshakes if five people shake hands with one

picture the

of handshakes

another.

NEW CONCEPTS Mixed-

We have been solving equations

number

4

coefficients

such as

= 7

5

by multiplying both sides of the equation by the reciprocal of the coefficient of x. Here the coefficient of x is |, so we multiply both sides by the reciprocal of |, which is f 1

l

3 x = —

t

t

1

1

4

—

simplified

35 x -

When

^

multiplied by f

-

7

equation that has a mixed-number coefficient, we convert the mixed number to an improper fraction as the first step. Then we multiply both sides by the solving

an

reciprocal of the improper fraction.

Lesson 90

Example

1

Solution

Solve:

First

3-x =

we

611

5

write 3| as an improper fraction.

—x 3

= 5

Then we multiply both the reciprocal of ~.

sides of the equation

by ^, which

is

l

i

y6

fraction form

X =

'

y&

^J

'

multiplied by

$

3 x = —

^

simplified

we

usually convert an improper fraction such as | to a mixed number. Recall that in algebra we usually leave improper fractions in fraction form. In arithmetic

Example

2

Solve:

l\y = l£ o

2

Solution

Since

we

fraction,

by a improper

will be multiplying both sides of the equation

we

first

convert both mixed numbers to

fractions.

15 —

5 -v y =

Then we multiply both

fraction

form

3

2

sides

by

11 13 | \y = f f 11 14 •

•

f

,

which

is

the reciprocal of

multiplied by §

Q

y = —

Negative coefficients

simplified

To solve an equation with

a negative coefficient,

we

multiply

both sides of the equation by a negative number. The coefficient of x in this equation is negative. (or divide)

-3x = 126

x 612

|

Saxon Math 8/7

solve this equation, we can either divide both sides by -3 or multiply both sides by -|. The effect of either method is ton make +1 the coefficient of x. We show both ways.

To

-3x = 126

-3x = 126 -3x

126 -3

-3

x = -42

x = -42 Example 3 Solution

Solve:

2

--x

=

4 —

We

multiply both sides of the equation by the reciprocal

-|,

which

of

is

2

"3 X

equation

—

V,i\

2

multiplied by —

3

X = Example 4 Solution

Solve:

simplified

-5x = 0.24

We may either multiply both by -5. Since the number, it appears

-5

by

—|

or divide both sides

right side of the equation is a decimal

that dividing

-5x = 0.24 -5x

sides

0.24

-5

x = -0.048

by -5 will be

easier.

equation

divided by -5 simplified

LESSON PRACTICE Practice set

Solve: a.

1-x = 36

b.

8

c.

2—w

3^a = 490 2

= 6-

4

d.

5

e.

-3x =

g.

-lOy = -1.6

0.45

12^y J = 5

3

r i.

h.

—m 3

2 = -

-2§w =

3|

613

Lesson 90

IXED PRACTICE

Problem set

1. (31.35)

2.

The sum

and 0.9 is how much greater than the and 0.9? Use words to write the answer.

of 0.8 o.8

p r0CU1 ct

f

For this

set of scores,

dnv.4) j

cj

mo(j e> anc

find the

mean,

(a)

3.

median,

range:

|

8, 6, 9, 10, 8, 7, 9, 10, 8, 10, 9,

(46)

(b)

8

The 24-ounce container is priced at $1.20. This container costs how much more per ounce than the 32-ounce container priced at $1.44?

4.

m

The

figure

right

at

is

a regular

decagon. One of the exterior angles is labeled a, and one of the interior angles is labeled b. (a)

What

the measure

is

of each

exterior angle of the decagon? (b)

5.

What

is

the measure of each interior angle?

Collect like terms:

x 2 + 2xy + y 2 + x 2 - y 2

(84)

Use 6.

m

ratio

The

boxes

to solve

sale price of

What was

problems 6 and

7.

$36 was 90 percent of the regular

price.

the regular price?

7. Seventy-five percent of the citizens voted for Graham. If m} there were 800 citizens, how many of them did not vote

for 8.

Graham?

Write equations to solve

(77)

9.

(a)

Twenty-four

(b)

Thirty

is

is

(a)

and

(b).

what percent of 30?

what percent

Use two unit multipliers

of 24?

to convert 2

2

ft

to square inches.

(88)

10. 1711

Diagram

this

statement.

Then answer

the questions

that follow.

Three hundred doctors recommended Brand X. This was | of the doctors surveyed. (a)

(b)

How many doctors were surveyed? How many doctors surveyed did Brand X?

not

recommend

614

Saxon Math 8/7

11. If

x =

4.5

and y = 2x +

1

,

then

y equals what number?

(41)

12.

Compare: aQ) ab

a


what

13. If the perimeter of a square is 1 foot,

m

14.

1

is

the area

oi

the square in square inches?

Complete the

table.

Decimal

Fraction

Percent

(48)

1.75

(a)

6%, what

15. If the sales-tax rate is (46)

is

(b)

the total price of a $325

printer including sales tax?

16. Multiply. Write the

product in scientific notation.

(83)

4

x 10 )(8 x 1CT

(6

17. (67,70)

A cereal box 8 inches long,

w^e

^

anc 12 inches j

(a)

What

is

(b)

What

is

the

tall is

volume

7 )

3

3 inches

in.

shown.

of the box?

12

in.

the surface area of the

box?

18. (a) (66,82)

Find the circumference of the circle at right.

(b)

Find the area of the

circle. Use 3.14

19. List the

whole numbers

for;r.

that are not counting numbers.

(86)

20.

m

The coordinates of three X(5,3),andr(5,0).

vertices of \Z\WXYZ are

(60)

(0, 3)

(a)

Find the coordinates of Z and draw the rectangle.

(b)

Rotate

CJWXYZ

90°

origin,

and draw

its

counterclockwise about image [ZWV'X'Y'Z'. Write

coordinates of the vertices. 21.

W

What mixed number

is

f of 20?

the tht

Lesson 90

22.

On

number

a

line

graph x


5

>

We

we

begin by divide both

inequality

+

5

added

5 to

both sides

2x > 6

simplified

—>-

divided both sides bv

2

1

2.

1

5>l

+

-

2

2

x >

3

simplified

We

check the solution by replacing x in the original inequality with numbers equal to and greater than 3. We try 3 and 4 below.

2x — 5 > 2(3)

2(4)

Procedures

-

-

1

original inequality

5

>

1

replaced x with 3

1

>

1

simplified and checked

5

>

1

replaced x with 4

3

>

1

simplified and checked

for solving inequalities

in a later course.

with a negative variable term will be taught

636

Saxon Math 8/7

Now we graph the —

i

3.

1-

1

1

x >

solution

-2-10

1

This graph indicates that

all

numbers

greater than or equal to

3 satisfy the original inequality.

LESSON PRACTICE Practice set*

Solve each equation.

Show

all steps.

a.

8x - 15 = 185

b.

0.2y + 1.5 = 3.7

c.

-m -1

d.

l\n + 3^ = 14

4

3

= 1 2

2

e.

-Bp + 36 = 12

g.

-~m

f.

+ 15 = 60

2

38 =

4w -

26

= 0.6d - 6.3

h. 4.5

Solve these inequalities and graph their solutions: i.

2x +

5

>

1

j.

2x -

5


-1

— 639

Lesson 93

24. (34)

What is number

sum

line

below?

of the

numbers labeled

I 1 1

*

A

and B on the

B

A

—

-«

the

1

*

1

1

1

1

1 1

0.4

0.5

Show each

For problems 25-28, solve the equation. 25.

3x +

2

= 9

26. (93)

(93)

27. 0.2y

-

1

—

1

= 7

28.

-w

+ 4 = 14

3

--m

(90)

(93)

step.

= -6

3

Simplify: 29. 3(2

3

+ Vl6) - 4° - 8

•

2

-3

(57, 63)

(b)

-3(4) + 2(3)

-

1

640

Saxon Math 8/7

LESSON

94

Compound

Probability

WARM-UP Facts Practice: Scientific Notation (Test S)

Mental Math: a.

(-144)

c.

5w

e.

6 x 2|

g.

25%

+

-r

(-6)

= 4.5

1.5

(1.5

d.

Convert 30°C to degrees Fahrenheit,

f.

of 40, x 4, +

2,

x 10~ 8 )(4 x 10 3 )

b.

10% more

-^6, x 9,

+

1,

than 50

f

,

x 3, +

1,

f

Problem Solving: Six identical blocks marked X, a 1.7-lb weight, and a 4.3-lb weight were balanced on a scale as shown. Write an equation to represent this balanced scale, and find the

weight of each block marked X.

NEW CONCEPT We know that the probability of getting heads on one toss of a coin is \. We can state this fact with the following notation in which P(H) stands

for "the probability of heads."

P(H) =

The

probability

\ of getting heads on the second toss of a coin is probability of getting two heads in a row

So the is | x | = |. We can illustrate this fact with a tree diagram. we toss a coin one time, we can get heads or tails. also

If

the

|.

first

toss

came up heads,

either heads or tails.

the second toss could

If

come up

641

Lesson 94

Likewise,

come up

From

if

the

first

came up

toss

tails,

the second toss could

either heads or tails.

the

tree

diagram,

we can

list

the

four

possible

outcomes:

HH

HT

TH

TT

outcomes is called a sample space. Since any one of the four outcomes in the sample space is equally likely, the probability of each outcome is one fourth. This

list

of

Thus, the probability of getting express this way: P{H, H) =

When

outcome

HH

is

|,

which we can

J

one

event does not affect the probability of a second event, the events are independent. Each coin toss is an independent event because the outcome of one toss does not affect the outcome of a subsequent toss. the

of

The probability of independent events occurring a specified order of each event.

the product of the probabilities

is

Thus, for example,

P[H, H, T) P{H,

T, T,

P[T, H,

Example

1

The

T,

H) H, T)

face of this spinner

in

is

divided into

four congruent sectors. What is the probability of getting a 2 on the first spin and a 1 on the second spin?

642

Saxon Math 8/7

Solution

probability of getting a 2 is \. The probability of getting a 1 is \. The probability of independent events occurring in a specified order is the product of the individual probabilities.

The

P{2, 1) =

Example 2

Jim tossed a coin once, and it turned up heads. What is the probability that he will get heads on the next toss of the coin?

Solution

Past coin tosses do not affect the probability of future coin tosses. So there are only two possible outcomes for the second toss. The coin will turn up either heads or tails. The probability that

Example 3

If

Solution

number

will turn

up heads

tossed once, what rolled will be 12?

a pair of dice

total

it

is

is \.

is

the probability that the

only one combination of two die faces that total 12, and that is 6 and 6. The probability of one die stopping its roll with 6 on top is |. So the probability of two dice stopping with 6 on top is

There

is

36

The sample space

below shows the 36

in the table

possible

combinations of rolling two dice. Since only one combination results in a total of 12, the probability of rolling 12 is

^. From

outcomes in the sample space, we see that the probability of rolling 11 is ^, which is ^, and the probability the

list

of

of rolling 10

which

is

is j?. 12'

Outcome •

•

•

•

•

of First Die

•

•

•

•

•

•

•

•

•

2

3

4

5

6

7

3

4

5

6

7

8

4

5

6

7

8

9

5

6

7

8

9

10

6

7

8

9

10

11

7

8

9

10

11

12

• •

•

• •

•

•

•

•

•

• •

•

•

• •

•

•

•

•

643

Lesson 94

Example 4

With one

toss of a pair of dice,

what

the probability of

is

rolling a total greater than 9?

Solution

Method

1:

From

the table showing sample space,

we

see that

there are 36 possible combinations. So 36

bottom term of the probability

the

is

Also from the table we see that six of the combinations total more than 9 (three total 10, two total 11, one totals 12). So 6 is the top term of the probability ratio.

P{>

Method

2:

9)

ratio.

=

We

regard each roll of a die as an independent event. We think about the combinations that total more than 9. If the first die stops on 1, 2, or 3, there is no way for the total to reach 10, 11, or 12. If the first die stops on 4, the second die must stop

on

We

6.

calculate this probability.

1

J

6 If

the

on

first

5 or 6

die stops

on

5,

the

on

the second die

first

die stops

4, 5, or 6 (three

on

6,

I.|

add these

stop

may

stop

the

total

A

favorable outcomes).

\ 6

probabilities

probability of rolling a

= P(>9) J 1

=

the second die

P(6, 4 or 5 or 6) =

We

may

(two favorable outcomes).

P(5,5or6) = If

36

6

number

A

= f 36 6

•

to

find

greater than

A + A + A = A 36 36 36 36

9.

= 1 6

When

the outcome of one event affects the probability of a subsequent event, the events are dependent. For instance, if a card is drawn from a deck and not replaced, the probabilities of

the draws of any remaining cards are affected

by the

first

draw.

The probability of dependent events occurring in a specified order is the product of the first event and the recalculated probabilities of each subsequent event.

Saxon Math 8/7

Example

5

From

deck of cards, Josefina selected and kept one card, then a second card, then a third card, and finally a a well-mixed

What

the probability that the four cards Josefina selected are aces? fourth card.

Solution

is

Since the drawn cards are not replaced, the events are dependent. So the probability of each event must be calculated as though the prior specified events had occurred.

Each card Josefina draws must be an ace. There are 4 chances out of 52 that the first card is an ace. That leaves 3 aces in 51 cards for the second draw. For the third draw there are 2 aces in 50 cards, and for the fourth draw there is 1 ace in 49 cards.

P{A, A, A, A) =

We show the

ill A A A £2 13

D

see® G2DCOCDGD

Many

•

calculation below.

P(A, A, A, A) =

c

— — — — 52 51 49 50

'

54 17

'

'

25

calculators have an exponent

that can be

used

i J_ = 49 270,725

key such

or

as

to calculate probabilities.

rasracD CD CD CD en CD CD CD CD CD CD CD CD CD CD GD CD

Suppose you were going

to take a ten-question true-false test.

Instead of reading the questions, you decide to guess every answer. What is the probability of guessing the correct answer to all ten questions?

The

probability of correctly guessing the

probability of correctly guessing the

The

first

answer is |. The two answers is 2

we may also write |) as 2 \ |2 = |, 2 4 probability of correctly guessing the first three answers

2'

2

first

or

|r.

\2)

'

Since \^^^

•

5

;

3

is

or ^. Thus, the probability of guessing the correct answer 10 or To find 2 10 on a calculator to all ten questions is (|) (|)

,

with a

B

,

or

aa key, we use these keystrokes:

Lesson 94

The number displayed

is

645

1024. Therefore, the probability of

correctly guessing all ten true-false answers

is

1

1024 Correctly guessing all ten answers has the same likelihood as tossing heads with a coin ten times in a row. Probabilities that are extremely unlikely

may

be displayed by

Find the probability of correctly guessing the correct answer to every question on a a calculator in scientific notation.

twenty-question, four-option multiple-choice

test.

SSON PRACTICE Practice set

a.

To win the game, Victor needs toss of a pair of dice.

do that on the

b.

Draw

What

is

with one the probability that he will to roll a total of 9

first try?

a tree diagram like the one at the beginning of the

lesson to find the sample space of three coin tosses.

c.

Jasmine is taking a four-option multiple-choice test. There are two answers she does not know. If she can correctly rule out one option on one question but no options on the other question, what is the probability she will correctly guess both answers?

d.

Quentin has a box containing a red marble, a white marble, and a blue marble. Quentin draws a marble, puts it back, and then draws again. Copy and complete the following table to show the sample space for two consecutive draws. The first row has been completed for you. ("R, W" means "red then white.") Second Draw Red 5

Red White

Blue

R,

R

White R,

W

Blue R,

B

646

Saxon Math 8/7

e.

Quentin draws from the box twice more. This time he di< not replace the marble after the first draw. Copy arn complete the following table to show the sample spao for these two draws. The first row has been completed fo you. Notice that the space for "red then red" is blank This outcome is not possible. If Quentin draws the rec

marble on the first draw and does not replace it, the marble will not be in the box for the second draw.

rec

Second Draw Red White Blue Red

R,

W

R,

B

CO

Q **

White

Blue

f.

Were

two

draws

problem d dependent or independent events? Were the two draws in problem e dependent or independent events? the

in

MIXED PRACTICE Problem set

1. 1511

2. (53,55)

Twenty-one billion Write the answer in

The

3.

4.

how much more

than 9.8 billion?

scientific notation.

an average speed of 48 miles per Yi 0UT for the first 2 hours and at 60 miles per hour for the next 4 hours. What was the train's average speed for the 6-hour trip? (Average speed equals total distance divided

by

l46)

is

train traveled at

total time.)

10-pound box of detergent costs $8.40. A 15-pound box costs $10.50. Which box costs the most per pound? How much more per pound does it cost?

A

In a rectangular prism,

what

is

the ratio of faces to edges?

(36, 67)

Use 5. (65)

ratio

boxes

to solve

problems 5-8.

The team's win-loss ratio was 3 to 2. If the team won 12 games and did not tie any games, how many games did the team play?

Lesson 94

6.

Twenty-four

is

The number

that

to 36 as

42

is to

6 47

what number?

(72)

7. {92)

8. (92)

9.

20%

is

what percent

of

were marked down 20 percent

to

than 360

less

is

360?

During the sale

shirts

$20. What was the regular price of the shirts (the price before the sale)?

Use two unit multipliers

perform each conversion:

to

(88)

10. (71)

(a)

12

(b)

1

2

ft

to square inches

kilometer to millimeters

Diagram

Then answer

statement.

this

the questions

that follow.

The duke conscripted two fifths of the male serfs in his dominion. He conscripted 120 male serfs in all. (a)

How many male

(b)

How many

serfs

male

were in the duke's dominion?

serfs

in his

dominion were not

conscripted?

11. If a pair of dice is tossed once, 1941

the total

(b)

y = 4x -

3

is

the probability that

rolled will be

1?

(a)

12. If

number

what

2?

(c)

3?

and x = -2, then y equals what number?

(91)

13. (16,20)

14.

The perimeter

square in square

Qf

The

of a certain square

sale price of the

(46,60)

g 5 p ercent

is

4 yards. Find the area

feet.

new

car

was $14,500. The

sales-tax

.

(a)

What was

the sales tax on the car?

(b)

What was

the total price including tax?

(c)

If

the commission paid to a salesperson is 2 percent of the sale price, how much is the commission on a

$14,500 sale?

648

Saxon Math 8/7

15.

Complete the

table.

Decimal

Fraction

(48)

66|%

(b)

(a)

(d)

(c)

16. (a)

Percent

What

is

200 percent of $7.50?

What

is

200 percent more than $7.50?

(60, 92)

(b)

product in scientific notation.

17. Multiply. Write the (83)

(2

8

x 10 )(8 x 10

box shown.

2 )

18. Robbie stores 1-inch cubes in a l70)

19. l82)

with inside dimensions as How many cubes will fit in this box?

The length

2

in.

6

8

14

of each side of the square

equals the diameter of the circle. area of the

square

is

in.

in.

in.

The

how much

greater than the area of the circle?

Use

20. Divide 7.2 l42 45) -

21. (85.In V .9)

22 —

by 0.11 and write the quotient with a bar over

therepetend.

Find the missing numbers in the by using mnction Then graph the function.

^

y = 3x X 3

-1

22. Solve this inequality

and graph

its

solution: 2x

(93)

measure of ZAOC is half the measure of ZAOD. The measure of ZAOB is one third the measure of ZAOD.

23. In the figure at right, the

m

for k.

(a)

FindmZ^OR

(b)

FindmZEOC.

2

y 1

1

1

1

U -

5

< -1

1

649

Lesson 94

24.

w

The length

BC

of segment

how much

is

C

B

\

1

1

1

1

1

1

I

I

1

|

I

I

|

|

I

I

I

+ 4 = 28

26. (90)

(93)

|

"II

3

For problems 25 and 26, solve the equation. 25. 1.2p

I I

|

2

1

I

I

|

|

inch

than the

AB?

length of segment

1

less

Show each

step.

-6§m = 1^ O 9

Simplify: 27.

(a)

6x 2 + 3x - 2x -

(b)

(5x)(3x)

1

(84, 87J

(-8)

28. (a) (85, 91)

(b)

_

l94)

(5x)(-4)

(-6)

-

(4)

-3 -5(-4) - 3(-2) - 1

29. Evaluate: b

30.

-

2

- 4ac

Hugo constructed

if

a = -1, b = -2, and c = 3

sample

space of the possible outcomes of flipping a quarter and a dime. He figured the quarter could land either heads up or tails up. For each a

quarter outcome he listed two outcomes for the flip of the dime. For one possible outcome, heads on both coins,

Hugo wrote

Qh Dh-

Complete Hugo's

possible outcomes ot the experiment.

list

to

show

all

650

Saxon Math 8/7

LESSON

95

Volume of a Right Solid

WARM-UP Facts Practice: Order of Operations (Test T)

Mental Math: + (-100)

a.

(72)

c.

60 Ton

e.

8 x 2|

g.

10%

y 1.5

x 10 )(2.5 x 10 6 ) 6

b.

(2.5

d.

Convert 25°C to degrees Fahrenheit,

50% more than 60 how much more than 20% of 100? f.

of 300

is

Problem Solving:

Copy

this

problem and

fill

in the missing digits:

_3

NEW CONCEPT A

right

solid

is

a

geometric

solid

whose

sides

are

perpendicular to the base. The volume of a right solid equals the area of the base times the height. This rectangular solid is a right solid. It is 5 long and 2 wide, so the area of the 2 base is 10

m

m

m

.

Z

/

/ /

One cube

will

fit

on each square meter of the base, and

cubes are stacked 3

m high, so

Volume =

area of the base x height

= 10 nr x

= 30

m

3

3

m

the

Lesson 95

651

the base of the solid is a polygon, the solid is called a prism. If the base of a right solid is a circle, the solid is called a right circular cylinder. If

Example

1

Right square

Right triangular

Right circular

prism

prism

cylinder

Find the volume of the right triangular prism below. Dimensions are in centimeters. We show two views of the prism.

Solution

The area

of the base

is

the area of the triangle. (4

Area of base =

The volume equals the

2

The diameter cylinder

What Solution

is

is its

of this

20 cm.

Its

„

= 6

cm 2

area of the base times the height.

Volume = Example

cm) (3 cm)

(6

cm 2 )(6 cm)

volume? Leave n

cm 3

circular

right

height

= 36

is

25 cm.

as n.

find the area of the base. The diameter of the circular base is 20 cm,

First

we

so the radius

is

10 cm.

2 2 Area of base = kt = ^(10 cm) = 100;r

The volume equals

cm 2

the area of the base times the height.

Volume = (100;rcm 2 )(25 cm) =

2500/r

cm 3

652

Saxon Math 8/7

LESSON PRACTICE Practice set

Find the volume of each right solid shown. Dimensions

are in

centimeters.

MIXED PRACTICE Problem set

1. t55)

2. (!nv 4)

The taxi ride cost $1.40 plus 35c for each tenth What was the average cost per mile for a 4-mile shows how many students earned certain scores on

The

table at right

the

last

test.

whisker plot

Create

box-and-

a

of a mile. taxi ride?

Class Test Scores

Score

for these scores.

Number of Students

100

95 90 85

im IH\

i

III

II

80 75

70

3. '

80)

The coordinates of the vertices of AABC are A (— 1, -1), B (-1, -4), and C(-3, -2). The reflection of AABC in the axis is its image AA'B'C Draw both triangles and write the coordinates of the vertices of AA'B'C '.

'.

Jackson is paid $6 per hour, 4 hours 20 minutes?

4. If '

53)

how much

will he earn in

653

Lesson 95

5. (36, 75)

what

In this rectangle,

Q£

t jie

shaded area

t

the ratio

is

unshaded

the

area?

600 pounds of sand costs $7.20, what would be the cost of 1 ton of sand at the same price per pound?

6. If (72)

7. 1921

The

cost of production rose

30%.

the

If

new

cost

is

$3.90

per unit, what was the old cost per unit? grocery store marks up cereal 30%, what is the retail price of a large box of cereal that costs the store $3.90?

8. If a {92)

9.

m 10. (71}

Use two unit multipliers

to convert

1000

mm

2

to square

centimeters.

Diagram

this

statement.

Then answer

the questions

that follow.

Three fifths of the Lilliputians believed in giants. The other 60 Lilliputians did not believe in giants.

11.

(a)

How many Lilliputians

(b)

How many Lilliputians believed in giants?

Compare: a

Qb

a is a counting an integer if

(79)

12. Evaluate:

m{m

were there?

+ n)

if

m

number and b

is

= -2 and n = -3

(91)

13. (94)

If

a pair of dice

the total (a)

14. 1951

is

number

7?

tossed once, what

the probability that

rolled will be (b)

a

number

Find the volume of the triangular prism shown. Dimensions are in millimeters.

is

less

than 7?

654

Saxon Math 8/7

15. l95)

16. (46)

The diameter of a soup can is 6 cm. Its height is 10 What is the volume of the soup can? (Use 3.14 for n.) Find the 3 tacos at

shake

including 6 percent sales tax, of $1.25 each, 2 soft drinks at 95(2 each, and a

total cost,

at $1.30.

17. Esther (94)

made

a tree diagram for tossing three coins. Copy

and complete this diagram, and then make a possible outcomes for the experiment. 1h)

i

—— —-

of

list

all the

HHH HHT

18. Simplify: (84 87) '

(a)

(-2xy)(-2x)(x 2 y)

(b)

6x - 4y +

3

- 6x - 5y -

19. Multiply. Write the

8

product in scientific notation.

(83)

(8

20. (a) (85,lnv.9)

x 10

_6 )(4

x 10 4 )

Find the missing numbers in by uging mnction rule. Then graph the function.

^

^

y = fx + X 6

(b)

At what point does the graph of the mnction intersect the y-axis?

4

1

y

n n

-2 i

21.

Find the measures of the following angles.

(40)

(a)

cm.

Zx

(c)

ZA

i

655

Lesson 95

22. 1781

23. (34)

On a number

line,

graph

all

the negative

numbers

greater than -2.

What is the average of the numbers the number line below? A

——

•

l

1 1

1

—B

f.

i

~#

1

1

+ 11 = 51

25. (93)

26. Solve this inequality

and graph

|x -

2

Show each = 14

3

its

solution:

(93)

0.9x + 1.2 < 3 Simplify: 27.

•

10

(57)

28.

Jr

*°

2

5

A

+ 2

- lO"1

+

(1

+ 2f

29. 5 (23, 26)

(63)

30. (a) (85,91)

(b)

(

- 10

—

1

1

»-

1.6

(93)

103

and B on

1

For problems 24 and 25, solve the equation.

-5w

A

labeled

1.5

24.

that are

7^-

(

-8 + 3(-2) - 6

- 6)

-

2=-

3

4

step.

a

656

Saxon Math 8/7

LESSON

96

Estimating Angle Measures Distributive Property with Algebraic Terms

WARM-UP Facts Practice: Two-Step Equations (Test U)

Mental Math: a.

(-27)

- (-50)

b.

c.

160 = 80 + 4y

e.

9 x if

g.

Estimate

x 10

5

)(2

x 10

7 )

d. Convert 15 °C to degrees Fahrenheit. f.

15%

(5

25% more

than $80

of $49.75.

Problem Solving:

What

is

the average of these fractions?

i

i t 4' 6' 12

NEW CONCEPTS Estimating angle

measures

We

have practiced reading the measure of an angle from a protractor scale. The ability to measure an angle with a protractor is an important skill. The ability to estimate an angle measure is also valuable. In this lesson we will learn a technique for estimating the measure of an angle. We will also practice using a protractor as we check our estimates.

measurement, we need a mental image of the units to be used in the measurement. To estimate angle measures, we need a mental image of a degree scale mental protractor. We can "build" a mental image of a the face of protractor from a mental image we already have

To estimate

a

—

—

a clock.

Lesson 96

The

6 57

which is 360°, and is divided into 12 numbered divisions that mark the hours. From one numbered division to the next is ^ of a full circle. One twelfth of 360° is 30 Thus the measure of the angle formed by the hands of a clock at 1 o'clock is 3010 at 2 o'clock is 60°, and at 3 o'clock is 90°. A clock face is further divided into 60 smaller divisions that mark the minutes. From one face of a clock

a full circle,

is

c

(

small division to the next 360° is 6°.

^

is

of a circle.

One

sixtieth of

60 )360° Thus, from one minute clock is 6°.

Here

mark

to the next

we have drawn an angle on the

of the angle set at 12,

is at

face of a clock.

the center of the clock.

and the other side of the angle

li

on the face of a

The vertex

One is

side of the angle is at "8 minutes after."

12

^10 -9

i

f

34

^

Since each minute of separation represents 6°, the measure of this angle is 8 x 6°, which is 48°. With some practice we can usually estimate the measure of an angle to within 5° of its actual measure.

Example

1

ZBOC in the

(a)

Estimate the measure of

(b)

Use

(c)

By how many degrees did your

a protractor to find the

measurement?

figure below.

measure of ZBOC. estimate differ from your

658

Saxon Math 8/7

Solution

(a)

We OC

use a mental image of a clock face on ZBOC with set at 12. Mentally we see that OB falls more than 10 minutes "after." Perhaps it is 12 minutes after. Since 12 x 6° = 72°, we estimate that mZBOC is 72°.

c

(b)

We trace ZBOC on we can

(c)

our paper and extend the sides so

use a protractor.

Our estimate

Make

a table of ordered pairs for the function

j]ieri g ra ph the function

16. Divide 6.75 l42)

y = x -

|,

on a coordinate plane.

by 81 and write the quotient rounded

to three

decimal places.

17. Multiply. Write the

product in scientific notation.

(83)

10

(4.8 x

18. Evaluate:

x 2 + bx +

_10

c

e

-6i x lCT )

)(6

if

x =

-3, b = -5,

and c =

6

(91)

19. (104)

Find the area of Dimensions are in Corners that look

this

figure.

millimeters.

square

are

square. (Use 3.14 for;r.)

20. 11051

Find the surface area of this right triangular prism. Dimensions are in centimeters.

21. (95)

23.

Find the volume of this right circular Dimensions are in inches. (Use 3.14 for;r.)

(a)

Solve for x: x + c = d

(b)

Solve for n: an = b

(106)

24. Solve: (102)

6w -

2(4 + w) =

w

+ 7

cylinder.

Lesson 106

25. Solve this inequality

and graph

its

741

solution:

(93)

6x + 8 < 14 26. Thirty-seven is five less than the product of (101)

what number

and three?

Simplify: 27. 25

-

[3

2

+ 2(5 -

3)]

(63)

6x 28. (103)

2

+ (5x)(2x)

4x

30. (_3)(-2)(+4)(-l) (103, 105)

_1 29. 4° + 3 (57)

+

(-3)

2

3

+ V^64 -

(-2)

3

+ 2

-2

742

Saxon Math 8/7

LESSON

107

Slope

WARM-UP Facts Practice: Percent-Decimal-Fraction Equivalents (Test Q)

Mental Math: a.

11000 (base

c.

(-2.5H-4)

e.

2x -

g.

75%

i.

U

+

3,

4.

-

DCCC

d.

(2.5

f.

2

of $60

7 x 8,

b.

1

=

2

2)

1,

h. -r

5,

3,

+

x 10

2

6 )

Convert -50°C to degrees Fahrenheit,

75% more than $60 2,

-r

5,

x

7,

+

1,

x 2,

-

1,

-r

3,

aT

Problem Solving:

we can find nine 1-by-l squares, four 2-by-2 squares, and

In the 3-by-3 square at right,

one 3-by-3 square. Find the total number of squares of any size in the 4-by-4 square.

NEW CONCEPT Below

are the graphs of

two functions. The graph of

the

function on the left indicates the number of feet that equal a given number of yards. Changing the number of yards by one changes the number of feet by three. The graph of the function on the right shows the inverse relationship, the number of yards that equal a given number of feet. Changing the number of feet by one changes the number of yards by one third. Yards to Feet

Feet to Yards

Number

of

Yards

Number

of Feet

Notice that the graph of the function on the left has a steep upward slant going from left to right, while the graph of the function on the right also has an upward slant but is not as

743

Lesson 107

steep.

The

slope.

We

"slant" of the graph of a function

assign a

number

called

is

to a slope to indicate

how

its

steep

and whether the slope is upward or downward. If the slope is upward, the number is positive. If the slope is downward, the number is negative. If the graph is horizontal, the slope

neither positive nor negative; it is zero. vertical, the slope cannot be determined.

the slope

graph

Example

1

State zero,

is

is

is

whether the slope of each or cannot be determined. y

(a)

If

the

line is positive, negative,

(b)

y=

5 - 4 -3 -2 -1

1

2

3

3

4

5

(d)

(c)

x = 3

5

Solution

To determine the

-4 -3 -2 -1_0

1

2

4

5

sign of the slope, follow the graph of the

function with your eyes from

left to right as

though you were

reading. (a)

From

left to right,

the graphed line rises, so the slope

is

positive. (b)

From

left to right,

the slope (c)

From

is

fall,

so

downward,

so

zero.

left to right,

the slope (d)

is

the graphed line does not rise or

the graphed line slopes

negative.

There is no left to right component of the graphed line, so we cannot determine if the line is rising or falling. The slope is not positive, not negative, and not zero. The slope of a vertical line cannot be determined.

744

Saxon Math 8/7

To determine the numerical value of the slope of a line, it is helpful to draw a right triangle using the background grid of the coordinate plane and a portion of the graphed line. First we look for points where the graphed line crosses intersections of the grid. We have circled some of these points on the graphs below.

we

two points from the graphed line and, following the background grid, sketch the legs of a right triangle so that the legs intersect the chosen points. (It is a helpful practice to first select the point to the left and draw the horizontal leg to the right. Then draw the vertical leg.) Next

select

y

y

We use the words

run and

rise to describe the

two

legs of the

the length of the horizontal leg, and the rise is the length of the vertical leg. We assign a positive sign to the rise if it goes up to meet the graphed line and a negative sign if it goes down to meet the graphed line. In the graph on the left, the run is 2 and the rise is +3. In the graph right triangle.

The run

is

Lesson 107

7 45

on the right, the run is 2 and the rise is -1. We use these numbers to write the slope of each graphed line. So the slopes of the graphed lines are these +3

rise

run

The slope ("rise

2

run

2

of a line

is

-1

rise

3

~

ratios:

2

the ratio of

1

~

2

its rise to its

run

over run"). slope =

rise

run

A line whose rise line

A

whose

rise

and run have equal values has a slope of 1. A has the opposite value of its run has a slope of —1.

than the lines above has a slope either greater than 1 or less than -1. A line that is less steep than the lines above has a slope that is between -1 and 1.

Example

2

line that is steeper

Find the slope of the graphed line below. y

-6 -5

-4 -3

-2 -1

6-5-' -

-

:

-. } -

:

-2

-3 —*t

1

;

j

--

: s

5

746

Saxon Math 8/7

Solution

note that the slope is positive. We locate and select two points where the graphed line passes through intersections of the grid. We choose the points (0, -1) and (3, 1). Starting from the point to the left, (0,-1), we draw the horizontal leg to the right. Then we draw the vertical leg up to (3, 1).

We

y

We

see that the run

is 3

and the

rise is positive 2.

We

write

the slope as "rise over run."

Slope =

|

Note that we could have chosen the points (-3, -3) and (3, 1). Had we done so, the run would be 6 and the rise 4. However, the slope would be the same because | reduces to |.

One way

check the calculation of a slope is to "zoom in" on the graph. When the horizontal change is one unit to the right, the vertical change will equal the slope. To illustrate this, we will zoom in on the square just below and to the right of the origin on this graph. to

The horizontal change is 1 unit to the right.

Lesson 107

Activity:

747

Slope

Materials needed: •

Photocopies of Activity Master 10 (1 each per student; masters available in the Saxon Math 8/7 Assessments and Classroom Masters)

Calculate the slope (rise over run) of each graphed line on the activity master

by drawing

right triangles.

-ESSON PRACTICE i

Practice set

a.

Find the slopes of the "Yards to Feet" and the "Feet Yards" graphs at the beginning of this lesson.

b.

Find the slopes of graphs

c.

Mentally calculate the slope of each graphed line below by counting the run and rise rather than by drawing right

(a)

and

(c)

in

example

to

1.

triangles. y

y

X

d.

For each unit of horizontal change to the right on the graphed lines above, what is the vertical change?

748

Saxon Math 8/7

MIXED PRACTICE Problem set

1. (92)

2. (51)

3.

The

shirt regularly priced at

What was

$21 was on sale

for |

off.

the sale price?

Nine hundred seventy-five billion is how much less than one trillion? Write the answer in scientific notation.

What

is

the

(a)

range and

(b)

mode

of this set of numbers?

(Inv. 4)

16, 6, 8, 17, 14, 16, 12

Use 4. {72]

5. l65)

6.

m

ratio

boxes

to solve

problems 4-6.

Riding her bike from home to the lake, Sonia averaged 18 miles per hour (per 60 minutes). If it took her 40 minutes to reach the lake, how far did she ride?

The

earthworms to cutworms in the garden was 5 to 2. If there were 140 earthworms and cutworms in the garden, how many were earthworms? ratio of

The average

cost of a

new

car increased 8 percent in one

year. Before the increase the average cost of a

was $16,550. What was the average

cost of a

new

new

car

car after

the increase?

7.

m

The points

(3,

-2), (-3, -2),

and

(-3, 6) are the vertices of

a right triangle. Find the perimeter of the triangle.

8. In this figure,

ZABC is a right angle.

(40)

(a)

Find

mZABD.

(b)

Find

mZDBC.

(c)

Find

mZBCD.

(d)

Which

triangles in this figure are similar?

Write equations to solve problems 9-11. 9.

Sixty

is

125 percent of what number?

(77)

10. Sixty is (77)

what percent of 25?

749

Lesson 107

11. Sixty is four

more than twice what number?

(101)

12. In a (94,

In, 10)

can are 100 marbles: 10 yellow, 20 red, 30 green, and

40blue (a)

If

a marble

is

drawn from the

can,

what

is

the chance

that the marble will not be red? (b)

13.

marble is not replaced and a second marble is drawn from the can, what is the probability that both marbles will be yellow? If

the

first

Complete the

table.

(48)

Fraction 5 6

14.

Compare:

(x

-

2

y)

(79)

O [y -

Decimal

Percent (b)

(a)

x]

2

if

x > y

15. Multiply. Write the product in scientific notation. (83)

(1.8 x

16. (a)

10 10 )(9 x 10" 6 )

Betw een which two consecutive whole numbers

is

(100, 105)

(b)

17.

What

are the

roots of 10?

y pairs for the function y = x + 1. T Graph these number pairs on a coordinate plane and

Find three

(Inv. 9, 107)

two square

x,

l

l

(a)

(b)

draw

a line through the points.

What

is

the slope of the graphed line?

cm, the area of the shaded

18. If the radius of this circle is 6 (104)

what

is

region?

Leave n as

of

this

19.

Find

11051

rectangular solid. Dimensions are in inches.

the

surface

area

n.

1

Saxon Math 8/7

20. 1951

Find

volume

the

circular cylinder.

of

right

this

Dimensions are

in centimeters. 21.

Find the

11051

cylinder in problem 20.

total surface area of the Use 3.14

22.

The polygon

ABCD is

a rectangle. Find

for;r

mZx.

(40)

23.

Find the slope of the graphed

line:

(107)

y

•j

4 3 2 -1

6-5-4-3-2-1 /

24. Solve for (106)

(a)

x

X 3

2

1

4

5

6

in each literal equation:

x - y = z

(b)

w

= xy

Solve: 25. (98)

— 21

=

—

26.

7

(102)

= 7 + 2x

6x +

5

5l +

3.5-1

Simplify: 27. 62

+ 5(20 -

2

[4

+ 3(2 -

1)]}

(63)

28.

^y

(6

(5)(-3)(2)(-4)

{0

_

(85)

I

-6

29.

+ (-2)(-3)

Lesson 108

751

LESSON

108

Formulas and Substitution

WARM-UP -x

Facts Practice: +

Algebraic Terms (Test V)

-f

Mental Math: a. c. e.

g. i.

1110 (base (-1)

5

+

2)

(-1)

5y - 2y = 24 150% of $120 At 60 mph,

b.

XLV

d.

(2.5

6

f.

h.

how far

5 3 x 10" )(4 x 10" )

Convert 3

sq.

yd

to sq.

ft.

$120 increased 50%

can Freddy drive in 3| hours?

Problem Solving:

m

Recall that \8 eans "the cube root of 8" and that Find Ail, 000,000.

a/8

equals

2.

NEW CONCEPT A

formula

a literal equation that describes a relationship

is

between two

more

or

variables.

Formulas are used in

mathematics, science, economics, the construction industry, food preparation anywhere that measurement is used.

—

To use

we

replace the letters in the formula with measures that are known. Then we solve the equation for the a formula,

measure

Example

1

Solution

we wish to

find.

Use the formula d =

rt to find

t

when d is

36 and r

is 9.

This formula describes the relationship between distance {d), rate (r), and time (f). We replace d with 36 and r with 9. Then we solve the equation for t.

d =

rt

36 = 9t t

Another way

to find

= 4 t is

formula substituted

divided by 9

to first solve the

formula for

d -

rt

formula

=

—

divided by r

t

r.

752

Saxon Math 8/7

Then replace d and

Example 2 Solution

r

with 36 and 9 and simplify.

t

=

t

= 4

—9

substituted

divided

Use the formula F = 1.8C + 32

to find

F when Cis

37.

This formula is used to convert measurements of temperature from degrees Celsius to degrees Fahrenheit. We replace C with 37 and simplify.

F

= 1.8C + 32

formula

F

= 1.8(37) + 32

substituted

F

= 66.6 + 32

multiplied

F =

added

98.6

Thus, 37 degrees Celsius equals 98.6 degrees Fahrenheit.

LESSON PRACTICE Practice set

a.

Use the formula

A

= bh

b.

Use the formula

A

= \bh

c.

Use the formula F = 1.8C + 32

to find

to find

b

when A

is

20 and h

is 4.

b

when A

is

20 and h

is 4.

to find

F when C is -40.

MIXED PRACTICE Problem set

1.

m

2. 1571

3. dnv. 4)

The main course

The beverage cost $1.25. Kordell left a tip that was 15 percent

cost $8.35.

Dessert cost $2.40. of the total price of the meal. Kordell leave for a tip?

How much money

Twelve hundred-thousandths is how much greater than twenty millionths? Write the answer in scientific notation. Arrange the following numbers in order from g rea t es t Then find the of numbers.

median and the mode

8, 12, 9, 15, 8, 10, 9, 8, 7,

4. (94)

did

Two

cards will be

drawn from

least to

of the set

4

a normal deck of 52 cards.

card will not be replaced before the second card drawn. What is the probability that both cards will be 5's?

The

first

is

753

Lesson 108

Use 5. (72)

6. (65)

7. (92>

ratio

boxes

to solve

problems 5-7.

300 Swiss francs. At that rate, dollars would a 240-franc Swiss watch cost?

Milton can exchange $200

how many

for

with red beans and brown beans in the ratio of 5 to 7. If there were 175 red beans in the jar, what was the total number of beans in the jar?

The

jar

was

filled

During the off-season the room rates at the resort were reduced by 35 percent. If the usual rates were $90 per day, what would be the cost of a 2-day stay during the offseason?

8.

Three eighths of a ton

is

how many pounds?

(60)

Write equations to solve problems 9—11. 9.

What number

is

800?

2.5 percent of

(60)

10.

Ten percent

of

what number

is

$2500?

(77)

11. Fifty-six is eight less than twice

what number?

(101)

12.

Find the slope of the graphed

line:

(107)

y

6 5

-4

-3 ^2

6

-5 -4 -3 -2 -

>

Lf

1

\3

X 4

5

6

-2 -3

-4 -5 -6

13. Liz is l98)

drawing a

equals 2 (a)

floor plan of her house.

On the plan,

1

inch

feet.

the floor area of a room that measures 6 inches by 7\ inches on the plan? Use a ratio box to solve the

What

is

problem. (b)

One

of the walls in Liz's

house

is

17 feet 9| inches

long. Estimate how long this wall would appear in Liz's floor plan, and explain how you arrived at your

estimate.

754

Saxon Math 8/7

14. ll01)

Find the measure of each angle of this triangle by writing and solving an equation.

15. Multiply. Write the

m

(2.8 x

16.

The formula

(108>

centimeters

17. dnv.9)

18. (104)

Make

(105)

c = 2.54n

(c).

10 5 )(8 x lCT 8 )

used to convert inches Find c when n is 12. is

[n) to

shows three pairs of numbers that satisfy function y = 2x. Then graph the number pairs on a coordinate plane, and draw a line through the points. a table that

Find the perimeter of this figure. Dimensions are in inches. (Use 3.14 for

19.

product in scientific notation.

5

K.)

Find the surface area of this cube. Dimensions are in inches.

10

10 10

20. (95)

Find

the

volume

circular cylinder.

of

this

right

Dimensions are

in centimeters. (Use 3.14 for;r.)

21.

Find

mZx in the figure below.

(40)

22.

These triangles are

similar.

Dimensions are in centimeters.

(a)

Find

(b)

Find the scale factor from the smaller

y.

to the larger

triangle. (c)

The

area of the larger triangle area of the smaller triangle?

is

how many

times the

Lesson 108

23. 1991

Use the Pythagorean theorem triangle from problem 22.

to find

x

755

in the smaller

24.

Find the surface area of a globe that has a diameter of

(105

10 inches. (Use 3.14 for

>

- x

25. Solve: l=-x = 32 3 (102)

Simplify: 26.

x 2 + x(x +

2)

(96)

29.

28. l.l{l.l[l. 1(1000)]}

30.

3^

(26)

(63)

(a)

(-6)

-

(7)(-4)

4

+ Vl25 + (-8X-9) (-3H-2)

(103, 105)

(b)

(-1)

+

(-1)

2

+ (-1)

3

+ (-1) 4

-f

10

756

Saxon Math 8/7

LESSON

109

Equations with Exponents

WARM-UP —

Facts Practice: +

x

Algebraic Terms (Test V)

Mental Math: a.

10010 (base

c.

HB) _

e.

g. i.

2.4 0.6

~

12|%

2)

b.

MCMLX

d.

(1.2

c f.

0.25

of $80 2,

1

^)

Convert 150

cm to

m.

%

less than $80 12f -l,f, x 4,-1,-r 3, square that

h.

Find | of 60, + 5, x number, - 1, 4- 2.

x lO

Problem Solving: Here are the front, top, and side views of an object. Draw a threedimensional view of the object from the perspective of the upper right front.

Front

Top

Right Side

NEW CONCEPT we have solved thus far, the variables have had an exponent of 1. You have not seen the exponent, because we usually do not write the exponent when it is 1. In this lesson we will consider equations that have variables In the equations

with exponents of

2,

such as the following equation:

3x z +

1

= 28

Isolating the variable in this equation takes three steps:

we

subtract 1 from both sides; next

then

we

we

first

divide both sides by

find the square root of both sides.

We show

results of each step below.

3x z +

1

= 28

3x 2 = 27

x2 =

9

x = 3,-3

equation subtracted 1 from both sides

divided both sides by

3

found the square root of both sides

3;

the

.

757

Lesson 109

Notice that there are two solutions, 3 and -3. Both solutions satisfy the equation, as we show below. 2

+

1

= 28

+

1

= 28

27 +

1

= 28

3(3)

3(9)

2

+

1

= 28

3(9)

+

1

= 28

27 +

1

= 28

3(-3)

28 = 28

When

the variable of an equation has an exponent of

remember Example

1

Solution

to look for

2

-

1

= 47

There are three

steps.

Solve: 3x

3x 2 -

= 47

1

Solution

Solve: 2x

We

2

4,

we

solutions.

We show the results added

x 2 = 16

x =

two

2,

of each step.

equation

3x 2 = 48

Example 2

28 = 28

1 to

both sides

divided both sides by 3

-4

found the square root of both sides

= 10

divide both sides by

2.

Then we

find the square root of

both sides.

2x 2 = 10

x2 =

equation

divided both sides by 2

5

x = V5 - V5 ,

found the square root of both sides

Since a/5 is an irrational number, we leave The negative of a/5 is -a/5 and not V-5

Example

3

Solution

Five less than what

We translate the

number squared

is

it

in radical form.

20?

question into an equation.

n2 -

5

= 20

We solve the equation in two steps. n2 -

5

= 20

n 2 = 25 77

= 5,-5

equation

added

5 to

both sides

found the square root of both sides

There are two numbers that answer the question, 5 and -5.

758

Saxon Math 8/7

Example 4

In this figure the area of the larger square

4 square units, which is twice the area of the smaller square. What is the length of each side of the smaller square?

is

Solution

We

will use the letter s to stand for the length of each side of 2 the smaller square. So s is the area of the small square. Since

the area of the large square (4) is twice the area of the small square, we can write this equation:

2s

We

2

= 4

solve the equation in two steps. 2

= 4

equation

2

- 2

divided both sides by 2

s

= il,

2s s

-A/2

found the square root of both sides

Although there are two solutions to the equation, there is only one answer to the question because lengths are positive, not negative. Thus, each side of the smaller square is V2 units.

Example

5

Solution

x

Solve: First

12

x

we

cross multiply.

Then we

find the square root of both

sides.

x

12

3

x

proportion

x z = 36

cross multiplied

x =6,-6

found the square root of both sides

There are two solutions

to the proportion, 6

and -6.

LESSON PRACTICE Practice set

Solve each equation: a.

3x 2 - 8 = 100

c.

Five less than twice what negative

b.

x 2 + x 2 = 12 number squared

is

157?

product of the square of a positive number and 21, then what is the number?

d. If the

e.

w — 4

=

—9 w

7

is

759

Lesson 109

MIXED PRACTICE Problem set

1. (45}

2. 11021

What

the quotient

is

sum

divided by the

when

of 0.2

the product of 0.2 and 0.05

and 0.05?

In the figure at right, a transversal

two

intersects

parallel lines.

(a)

Which

(b)

Which

angle is the interior angle to Zd?

alternate

(c)

Which

alternate

angle corresponds to

angle

is

(d)

the measure of

If

the

Zd?

Zb?

exterior angle to

Za

is

m and the measure of Zb

then each obtuse angle measures 3. 1101

4.

is

Twenty

more than decimal number? fiye

is

Use two unit multipliers

to

how many

degrees?

what

the product of ten and

conyert

1

km

2

to

3m.

is

square meters.

(88)

5. (36)

Use 6. "

7.

What

Santiago has S5 in quarters and S5 in dimes. ratio of the

number

of quarters to the

boxes

to solye

problems 6-8.

ratio

Jaime ran the how long will Sixty

is

number

of

is

dimes?

3000 meters in 9 minutes. At that take Jaime to run 5000 meters?

first it

the

rate.

20 percent more than what number?

(92)

8. 192

To

attract

customers, the merchant reduced

an equation - Write Sixty

{71)

prices

by

25 percent. What was the reduced price of an item that cost S36 before the price reduction?

9.

10.

all

Diagram

this

is

to solye this

problem:

150 percent of what number?

statement.

Then answer

the questions

that follow.

Diane kept | of her baseball cards and gave the remaining 234 cards to her brother. (a)

How many to

(b)

cards did Diane haye before she gave

her brother?

How many baseball

cards did Diane keep?

some

760

Saxon Math 8/7

11.

O

Compare: a - b

b - a

a > b

if

(79)

12. (94)

13. (75)

Warner knew the

answer

correct

20

to 15 of the

true-false

questions but guessed on the rest. What is the probability of Warner correctly guessing the answers to all of the remaining true-false questions?

Find the area of this trapezoid. Dimensions are in centimeters. h. 13

14. (95}

15. (105)

Find the volume of this triangular prism. Dimensions are in inches.

A

rectangular

label

is

wrapped

h-

— — 6

around a can with the dimensions shown. The label has an area of how

many

3

square inches? Use 3.14

16.

-I

in.

The skateboard

costs $36.

The

tax rate

is

for;r.

6.5 percent.

(46)

17.

(a)

What

is

the tax on the skateboard?

(b)

What

is

the total price, including tax?

Complete the

table.

What number

is

lo/

(b)

(a)

18.

Percent

Decimal

Fraction

(48)

/o

2

66| percent more than 48?

(92)

19. Multiply. Write the

product in scientific notation.

(83)

(6

20. (Inv.9,107)

8

x icr )(8 x 10

Find the missing numbers in the by using mnct i on e Then graph the function on a

m

coordinate plane. What of the graphed line?

is

4 )

.

.

.

2 3

x -

l

the slope

X

y

6

n i

i

i

i

-3

in.

)

Lesson 109

21. (65)

Use

box

solve this problem. The ratio of the measures of the two acute angles of the right ratio

a

triangle

7

is

to

to

What

8.

is

the

measure of the smallest angle of the triangle? 22.

The

(101)

measures of four central angles of a

shown

circle is is

l99)

in this figure.

the

What

measure of the smallest

the

central angle 23.

between

relationship

shown?

We can use the Pythagorean theorem to

find the distance between

points on a coordinate plane. the distance from point

we draw

To

two

M

find

M to point P, -5 -4 -3 -2 -1

a right triangle and use the

lengths of the legs to find the length

hypotenuse. What is the distance from point Mto point P?

of

the

Solve: 24.

3m 2

+

2

= 50

25. 7(y

-

= 4 - 2y

2)

(102)

(109)

Simplify: 26.

Vl44 - (V36)(V5)

27. (96)

(20)

28. fl|\l.5) (43)

x 2y + xy 2 + x{xy -

V

-r

9/

(-18) 30. (a) (57, 91, 105)

+ (-12) -3

(b)

^/1000

(c)

2

2

+

2

29. 9.5

2| O

(43)

(-6)(3)

- ^125 1

1 + 2° + 2"

- [4- V

5

3.4 J

y'

762

Saxon Math 8/7

LESSON

110

Simple Interest and Compound Interest

Successive Discounts

•

WARM-UP Facts Practice: Percent-Decimal-Fraction Equivalents (Test Q)

Mental Math: a.

11110 (base

2)

c. e.

4w -1

8-

150%

i.

Start

= 9

b.

DCLXXVIII

d.

(9 x

f.

10

6

)(6

x 10

9 )

Convert 1.5 L to mL.

$60 increased 50% with the number of minutes in half an hour. Multiply by of $60

h.

add the number of years in a decade; then find the square root of that number. What is the

number

the

of feet in a yard;

answer?

Problem Solving: At nine o'clock the hands of a clock form a 90° angle. What angle is formed by the hands of a clock l| hours after nine o'clock?

NEW CONCEPTS Simple interest

and

When you

hold your money

compound money interest

deposit other

in

money

in a bank, the

bank does not simply

spends your make more money. For this

for safekeeping. Instead,

places

to

it

opportunity the bank pays you a percentage of the money deposited. principal.

The amount of money you deposit is called the The amount of money the bank pays you is called

interest.

between simple interest and compound interest. Simple interest is paid on the principal only and not paid on any accumulated interest. For instance, if you deposited $100 in an account that pays 6% simple interest, you would be paid 6% of $100 ($6) each year your $100 was on deposit. If you take your money out after three years, you There

is

a difference

would have

a total of $118.

Lesson 110

763

Simple Interest $100.00

+

principal

$6.00

first-year interest

$6.00 $6.00

second-year interest

$118.00

third-year interest total

accounts, however, are compoundinterest accounts, not simple-interest accounts. In a compound-interest account, interest is paid on accumulated interest as well as on the principal. If you deposited $100 in an account with 6% annual percentage rate, the amount of interest you would be paid each year increases if the earned interest is left in the account. After three years you would have a total of $119.10.

Most

interest-bearing

Compound

Interest

$100.00 $6.00

principal

$106.00 $6.36

total after

$112.36 $6.74

total after

$119.10

total after three years

first-year interest

(6%

of $100.00)

one year

second-year interest (6% of $106.00)

two years third-year interest (6% of $112.36)

Notice that in three years, $100.00 grows to $118.00 at 6% simple interest, while it grows to $119.10 at 6% compound interest. The difference is not very large in three years, but as this table shows, the difference can become large over time. Total Value of $100 at

1

Make

Simple

Compound

Years

Interest

Interest

3

$118.00

$119.10

10

$160.00

$179.08

20

$220.00

$320.71

30

$280.00

$574.35

40

$340.00

$1028.57

50

$400.00

$1842.02

$1000 investment 10% compounded annually after 1, 2, 3, 4, and

a table that

growing 5 years.

at

Interest

Number of

Example

6%

shows the value

of a

764

Saxon Math 8/7

Solution

After the

first

year,

$1000 grows 10%

to $1100. After the

second year, the value increases 10% of $1100 ($110) to a total of $1210. We continue the pattern for five years in the table below. Total Value of $1000 at

Number

of

Years

10%

Interest

Compound

Interest

1

$1100.00

2

$1210.00

3

$1331.00

4

$1464.10

5

$1610.51

Notice that the amount of money in the account after one year is 110% of the original deposit of $1000. This 110% is composed of the starting amount, 100%, plus 10%, which is the interest earned in one year. Likewise, the amount of money in the account the second year is 110% of the amount in the account after one year. To find the amount of money in the account each year, we multiply the previous year's balance by 110% (or the decimal equivalent, which is 1.1). SAXON i

)

G3CDGDCD aaeo CD GD CD CD CD CD CD CD CD CD CD CD CD CD GD CD

Even with a simple calculator we can calculate compound interest. To perform the calculation in example 1, we could follow this sequence:

1000 x

The

circuitry

of

1.1

x 1.1 x 1.1 x 1.1 x 1.1

some

calculators

permits

= repeating

a

calculation by pressing the IB key repeatedly. To make the calculations in example 1, we try this keystroke sequence: +

This keystroke sequence

first

enters 1.1,

which

is

the decimal

form of 110% (100% principal plus 10% interest), then the times sign, then 1000 for the $1000 investment. Pressing the Bi key once displays

(

+

nn

f

f

'

'uu.

I

This calculator function varies with make and model of calculator. See instructions for your calculator if the keystroke sequence described in this lesson does not work for you.

765

Lesson 110

which

B

the value ($1100) after one year. Pressing the key a second time multiplies the displayed number by 1.1, the first number we entered. The new number displayed is is

/ 1

3 1-

I (

n U.

representing $1210, the value after two years. Each time the ^9 key is pressed, the calculator displays the account value after a successive year. Using this method, find the value of the account after 10 years and after 20 years.

Try entering the factors in the reverse order.

Are the same amounts displayed as were displayed with the prior entry when the IB key is repeatedly pressed? Why or

Example 2

why

not?

Use

a calculator to find the value after 12 years of a

investment that earns Solution

The

7|%

interest

7|%, which

compounded

$2000

annually.

0.075 in decimal form. We want to find the total value, including the principal. So we multiply the $2000 investment by 107|%, which we enter as 1.075. The keystroke sequence is

We

interest rate is

We

2

17. Multiply. Write the product in scientific notation. (83)

(6.3

18. Solve for y:

\y - x +

7

x 10 )(9 x 1(T 3 ) 2

(96, 106)

19.

What

lno)

of $4000 at

20.

The

is

the total account value after 3 years on a deposit

9%

triangles

interest

below

compounded annually?

are similar.

Dimensions are in inches.

(97, 98)

(a)

Estimate, then calculate, the length x.

(b)

Find the scale factor from the larger

to the smaller

triangle.

21. Find the l95)

volume

of this triangular

prism. Dimensions are in inches.

22.

Find the

11051

triangular prism in

23.

Find

total surface area of the

problem

21.

mZx in the figure at right.

(40)

Solve:

18

24. YL 2 (109)

25.

w

(109)

3-w 2 -

4 = 26

3

Simplify:

-

26. 16

(27

-

3 [8

-

2

(3

-

2

3

)]}

(63)

2

[6ab ){8ab) 27. (W3)

12a b

29. 20 (26)

28.

2u2

+

3^

\ O

(43)

-r

1O

3^ + 1.5 + 4|

30. (-3) (103)

O

O

2

+ (-2)

:

770

Saxon Math 8/7

Focus on

Scale Factor in Surface Area

and Volume In this investigation

we

will study the relationship between

and volume of three-dimensional begin by comparing the measures of cubes of

length,

surface

shapes.

We

area,

different sizes.

Activity: Scale Factor in Surface

Area and Volume

Materials needed by each group of 2 to 4 students:

photocopies of Activity Master 11 (available in Saxon Math 8/7 Assessments and Classroom Masters) or 3 sheets of 1-cm grid paper

• 3

•

Scissors

•

Tape

Use the materials to build models of four cubes with edges 1 cm, 2 cm, 3 cm, and 4 cm long. Mark, cut, fold, and tape the grid paper so that the grid is visible when each model is finished.

One

pattern that folds to form a model of a cube below. Several other patterns also work.

is

shown

i

Investigation 11

Copy

this table

on your paper and record the measures

771

for

each cube. Measures

Four Cubes

2-cm cube

1-cm cube

Edge

of

3-cm cube

4-cm cube

length (cm)

Surface area (cm 2 )

Volume (cm 3 )

Refer to the table to answer the following questions:

Compare 1.

the 2-cm cube to the 1-cm cube.

The edge length

of the

2-cm cube

is

how many

times the

is

how many

times the

edge length of the 1-cm cube? 2.

The surface area

of the

2-cm cube

surface area of the 1-cm cube?

3.

The volume of the 2-cm cube volume of the 1-cm cube?

Compare the 4-cm cube 4.

The edge length

to the

of the

is

how many

times the

2-cm cube.

4-cm cube

is

how many

times the

is

how many

times the

edge length of the 2-cm cube? 5.

6.

4-cm cube surface area of the 2-cm cube?

The surface area

of the

The volume of the 4-cm cube volume of the 2-cm cube?

Compare the 3-cm cube 7.

The edge length

to the

of the

is

how many

times the

1-cm cube.

3-cm cube

is

how many

times the

is

how many

times the

edge length of the 1-cm cube? 8.

The surface area

of the

3-cm cube

surface area of the 1-cm cube?

772

Saxon Math 8/7

how many

times the

Use the patterns that can be found in answers 1-9 the comparison of a 6-cm cube to a 2-cm cube.

to predict

is

how many

times the

is

how many

times the

9.

10.

The volume of the 3-cm cube volume of the 1-cm cube?

The edge length

of a

is

6-cm cube

edge length of a 2-cm cube?

11.

The surface area

of a

6-cm cube

surface area of a 2-cm cube?

12.

The volume of a 6-cm cube volume of a 2-cm cube?

13. Calculate

(a)

is

the surface area and

how many

(b)

the

times the

volume

of a 6-cm

cube.

14.

The calculated

6-cm cube

surface area of a

is

how many

times the surface area of a 2-cm cube?

15.

The calculated volume of a 6-cm cube the volume of a 2-cm cube?

In problems 1-6

is

we compared the measures

how many times

2-cm cube to a 1-cm cube and the measures of a 4-cm cube to a 2-cm cube. In both sets of comparisons, the scale factors from the smaller cube to the larger cube were calculated.

Scale Factors from Smaller

Cube

to Larger

Measurement Edge

Cube

Scale Factor 2

length

Surface area

Volume

2

2

= 4

23 = 8

of a

Investigation 11

we compared

Likewise, in problems 7-15

3-cm cube

to a

1-cm cube and

a

6-cm cube

773

the measures of a to a

2-cm cube.

We

calculated the following scale factors:

Scale Factors from Smaller

Cube

to Larger

Measurement Edge

Cube

Scale Factor

length

Surface area

Volume

3

3

3

2

3

= 9

= 27

Refer to the above description of scale factors to answer problems 16-20. 16.

Manuel calculated the scale factors from a 6-cm cube to a 24-cm cube. From the smaller cube to the larger cube, what are the scale factors for (a) edge length, (b) surface area, and (c) volume?

17.

Bethany noticed that the scale factor relationships for cubes also applies to spheres. She found the approximate diameters of a table tennis ball (l|

and

a

playground ball

(9 in.).

in.),

a baseball (3

Find the scale

in.),

factor for

volume of the table tennis ball to the volume of the baseball and (b) the surface area of the baseball to the (a)

the

surface area of the playground ball.

18.

from 2|-by3§-in. wallet-size photos. Find the scale factor from the smaller photo to the enlargement for (a) side length and

The photo

(b)

19.

lab

makes

5-by-7-in. enlargements

picture area.

Rommy wanted

to charge the

same price per square inch

of cheese pizza regardless of the size of the pizza. Since all of Rommy's pizzas were the same thickness, he based his prices on scale factor for area. If he sells a 10-inch

diameter cheese pizza for S10.00, how much should he charge for a 15-inch diameter cheese pizza?

774

Saxon Math 8/7

20.

The Egyptian

knew

archaeologist

that

the

scale-factoi

relationships for cubes also applies to similar pyramids.

The

~

model of the Great Pyramid. Each edge of the base of the model was 2.3 meters, while each edge of the base of the Great Pyramid measured 230 meters. From the smaller model to the Great Pyramid, what was the scale factor for (a) the length of corresponding edges, (b) the area of corresponding faces, and (c) the volume of the pyramids? archaeologist built a

scale

Notice from the chart that you completed near the beginning of this investigation that as the size of the cube becomes

and volume become much greater. Also notice that the volume increases at a faster rate than the surface area. The ratio of surface area to volume changes as

greater, the surface area

the size of an object changes.

Ratio of Surface Area to

1-cm cube

Volume

of Four

2-cm cube

3-cm cube

3 to

2

Cubes 4-cm cube

Surface Area to

6

to

1

1

to

1

1.5 to

1

Volume

The

ratio of surface area to

volume

affects the size

of containers used to package products.

The

and shape

ratio of surface

volume also affects the world of nature. Consider the relationship between surface area and volume as you answer area to

problems 21-25.

1-cm were cubes arranged to form one large cube. Austin wrapped the large cube with paper and sent the package to Betsy. The volume of the package was 64 cm 3 What was the surface area of the exposed wrapping

21. Sixty-four

.

paper?

Investigation 11

22.

775

When

Betsy received the package she divided the contents into eight smaller cubes composed of eight 2-cm cubes. Betsy wrapped the eight packages and sent them on

The

total

eight packages

was

to Charlie.

volume still

64

of the

cm 3

.

What was

the total surface area of the exposed wrapping paper of the eight packages?

opened each of the eight packages and wrapped each 1-cm cube. Since there were 64 cubes, the total volume was still 64 cm 3 What was the total surface area of exposed wrapping paper for all 64 packages?

23. Charlie

.

summer

picnic the ice in two large insulated containers was emptied on the ground to melt. A large block of ice in the form of a 6-inch cube fell out of one container. An equal quantity of ice, but in the form of 1-inch cubes, fell scattered out of the other container. Which, if either, do you think will melt sooner, the large block of ice or the small scattered cubes? Explain

24. After a

your answer. 25.

much, we might say that he or she "eats like a bird." However, birds must eat large amounts, relative to their body weights, in order to maintain their body temperature. Since mammals and birds regulate their own body temperature, there is a limit to how small a mammal or bird may be. Comparing a hawk and a sparrow in the same environment, which of the two If

someone does not

eat very

might eat a greater percentage of day? Explain your answer. a.

Investigate

how the

its

weight in food every

weight of a bird and

its

wingspan

are

related.

reasons why the largest sea mammals are so larger than the largest land mammals.

b. Investigate

much c.

25% taller than Brad and weighs twice as much. Explain why you think this height-weight relationship may or may not be reasonable. Brad's dad

is

77*,

;^/'.- v*** *

7

LESSON

111

Dividing in Scientific Notation

WARM-UP

—

Facts Practice:

- Als^c-r^ic Terms (Test Vj

Mental Math: b.

CCCXXI

f-5)

d.

[8

= 600

f

Convert

a.

10111(base2J

c.

f-0.25)

3m 'j

g.

3

;

4

7/77

%

of SI 50

Estimate

i.

8%

.

on

2

1 ft

2

to square inches.

S150 reduced by 33f %

h.

tax

x 10" 4)

a Si 98. 75 purchase.

Problem Solving: Carpeting is sold by the square yard. If carpet is priced at S25 per square yard (including tax and installation), how much would it cost to carpet a classroom that is 36 feet long and 36 feet wide?

NEW CONCEPT One

unit astronomers use to

measure distances within

solar system is the astronomical unit (AU).

unit

is

which

An

our

astronomical

the average distance between Earth and the Sun, is

roughly 150,000,000

km

(or

astronomical unit (AU)

1

about 150,000,000

93,000,000 mi).

Sun

"

km

Earth

For instance, at a point in Saturn's orbit when it is kilometers from the Sun, its distance from the Sun 1,500,000,000

km

1

AU

150,000,000

km

= 10

1.5 billion is

10 AU.

AU

This moans that the distance from Saturn to the Sun is about limes the average distance between Earth and the Sun. 1

When

dividing very large or very small numbers, it is helpful to use scientific notation. Here we show the same calculation in scientific

notation:

777

Lesson 111

we

In this lesson

will practice dividing

numbers

in scientific

notation.

we

when we

multiply numbers in scientific notation, multiply the powers of 10 by adding their exponents.

Recall that

(6

x 10 6 )(1.5 x 10 2 )

we have

Furthermore,

this

= 9 x 10 8

important rule:

When we

divide numbers written in scientific notation, we divide the powers of 10 by subtracting their exponents.

6 x 10 1.5

Example

1

10'

^

(6

"

=

2

4)

Write each quotient in scientific notation: ,

6 x 10

,

8

(a)

1.2 x 10

Solution

x

= 4 x 10 4

(a)

To

3 x 10

fU (b)

r 6

f

b

The quotient (b)

We

is

10

8

2 10 6 = 10

-f

5 x 10

The

-r

8

*~

(8

~ 6 "

.

2J

2 .

6 .

0.5

10 3

—

8 6 divide 6 by 1.2 and 10 by 10

we

3 divide 3 by 6 and 10 by 10

613^

2 2 x 10"

8 x 10

5.

12.}6uT

,

(c)

6 x 10

find the quotient,

3

3 10 6 = 10~

3 quotient, 0.5 x 10~

is

,

—

- 6 = -3)

not in proper form.

We write

the quotient in scientific notation. 5 x 1(T (c)

We

4

8 2 divide 2 by 8 and 10~ by 10"

8)2^00

,-2 2 10"

.

"T

.

n -8a _ 10" 10' = in6 ,

—

[-2 - (-8) = 6]

6 is not in proper form. quotient, 0.25 x 10 write the quotient in scientific notation.

The

,

2.5 x 10

Example 2

We

5

8 about 1.5 x 10 km. 5 Light travels at a speed of about 3 x 10 km per second. About how many seconds does it take light to travel from the

The distance from the Sun

Sun

to Earth?

to Earth is

778

Saxon Math 8/7

Solution

We

8 divide 1.5 x 10

km

by

3 x 10

1.5x10" km 3 x

We

may

notation, 5 x 10

2

s.

km/s.

=0 5xl0 3 s

5

10 km/s the

write

5

quotient

We may

proper

in

write

also

the

scientific

answer

in

standard form, 500 s. It takes about 500 seconds for light from the Sun to reach Earth.

LESSON PRACTICE Practice set*

Write each quotient in scientific notation: 3.6 x 10

9

7.5 x 10

,

3

d.

a.

2 x 10

3

2.5 x 10

8 4.5 x 10"

,

4 3 x 10~

4 x 10

4 6 x 10~ 8 1.5 x 10"

12

e.

9

r

1.5 x 10

4

i.

8 x 10

4

3 x

8 3.6 x 10~

10

12

2 1.8 x 10"

,

2 6 x 10"

9 x 10

-8

MIXED PRACTICE Problem set

1. (12)

2. (28)

3. (36, 54)

Indian-head penny was minted in 1859. The last Indian-head penny was minted in 1909. For how many years were Indian-head pennies minted?

The

first

The product and 15?

of

y and 15

is

600.

What

is

the

sum

of

y

Thirty percent of those gathered agreed that the king sh^id abdicate his throne. All the rest disagreed. (a)

What

(b)

What was

fraction of those gathered disagreed?

the ratio of those

who

agreed to those

who

disagreed? 4.

m

Triangle ABC with vertices A (0, 3), B (0, 0), and C (4, 0) is translated one unit left, one unit down to make the image kA'B'C. What are the coordinates of the vertices of

AA'B*C?

779

Lesson 111

5.

(a)

Write the prime factorization of 1024 using exponents.

(b)

Find V1024.

(21)

6. l89)

A

portion of a regular polygon is shown at right. Each interior angle

150°

measures 150°. (a)

What

(b)

The polygon has how many

(c)

What

is

the measure of each exterior angle?

the

is

name

for a

sides?

polygon with

this

number

of

sides? 7. (92)

8. 1941

9.

The sale price of an item on was the regular price?

sale for

40%

In a bag are 12 marbles: 3 red, 4 white,

off is $48.

and

What

5 blue.

One

marble is drawn from the bag and not replaced. A second marble is drawn and not replaced. Then a third marble is drawn. (a)

What

is

(b)

What

is

the probability of drawing a red, a white, and a blue marble in that order?

the probability of drawing a blue, a white, and a red marble in that order?

Write an equation to solve this problem:

(101)

Six more than twice what 10.

What

tl01)

angle of this triangle?

is

number

is

36?

the measure of each acute

11. Solve fore

2 :

c

2

- b2 = a2

(106)

12. In the figure below, (102)

if 1

II

q and

mZh

= 105°, what

the measure of

(a)

Za?

(b)

Zb?

(c)

Zc?

(d)

Zd?

is

780

Saxon Math 8/7

13. tW8)

The formula below may be used to convert temperature measurements from degrees Celsius (C) to degrees Fahrenheit (F). Find Fto the nearest degree when Cis 17.

F 14. 11041

What

is

= 1.8C + 32

the area of a 45° sector of a

with a radius of 12 in.? Use 3.14 for k and round the answer to

circle

the nearest square inch.

15. nv g) '

Make

showing three or four the equation x + y = 1. Then graph all

a table of ordered pairs

solutions for

possible solutions. 16. Refer to the

graph in problem 15

to

answer

(a)

and

(b).

(W7)

17. ll05)

(a)

What

(b)

Where does the graph of x + y =

is

the slope of the graph of

x + y =

1?

1 intersect the y-axis?

The students in Room 8 decided to wrap posters around school trash encourage cans to properly dispose of illustration

others trash.

to

The 36

shows the dimensions

in.

of the trash can. Converting the

dimensions

to feet

for k, find the

and using 3.14

number

paper needed around each trash can. of

feet

18. (95)

19.

The to

20. ("»

hold

how many

2x 2 +

What

is

1

wrap

problem 17 has the capacity

cubic feet of trash? to

each of these equations:

= 19

(b)

2x 2 -

1

= 19

the perimeter of a triangle with vertices (-1,

2),

(-i,-i),and(3,-l)?

21. Sal deposited (110)

to

trash can illustrated in

Find two solutions (a)

of square

compounded

$5000 in an account

annually. account after 5 years?

What was

that paid

5%

interest

the total value of the

Lesson 111

22. (97,991

The

figure

right

at

similar triangles.

BC is

and

If

shows three

AC

is

15

cm

20 cm,

(a)

what

is

AB?

(b)

what

is

CD?

each quotient in scientific notation:

23. Express 11111

„„-8 n 3.6 x 10'

3.6 x 10 (b)

(a)

6 x 10

(

1.2 x 10

24. In the figure below, if the

m

the measure of

measure of

Zx

-6

is

140°,

Ay?

Solve: 25.

5x + 3x = 18 + 2x

26. (98)

(102)

3.6

4.5

X

0.06

Simplify: 27.

(a)

(-1)

6

+ (-1)

;

6

(b)

(-10)

(b)

x(x -

* (-10)

(103)

2

2

28.

(4a b){9ab c) (a)

6abc

(96, 103)

29. (-3) + (+2)(-4) (85)

3|

30. -3 (43,45)

.

l| + 1.5 *

0.03

- (-6H-2) -

(-8)

c)

+ cx

what

782

Saxon Math 8/7

LESSON

112

Applications of the

Pythagorean Theorem WARM-UP Facts Practice: Multiplying and Dividing in Scientific Notation (Test

W)

Mental Math: a.

100000 (base 2

2)

+ (-10)

3

c.

(-10)

e.

m

g.

25%

of $2000

i.

Start

with 2 dozen, +

2

AT,

= 100

-r

b.

XCIX

d.

(8

f

Convert 50°C to degrees Fahrenheit,

.

h. 1,

x 10 6

)

-r

(4

x 10

$2000 increased x 4, + 20,

-r

3,

3 )

25%

+

~-

2,

6,

x 4,

-

3,

2.

Problem Solving: Mariabella was | of the way through her book. Twenty pages later she was | of the way through her book. When she is | of the way through the book, how many pages will she have to read to finish the book?

NEW CONCEPT Workers who construct buildings need to be sure that the structures have square corners. If the corner of a 40-foot-long building is 89° or 91° instead of 90°, the other end of the building will be about 8 inches out of position.

One way

construction workers can check whether a building under construction is square is by using a Pythagorean triplet. The numbers 3, 4, and 5 satisfy the Pythagorean

theorem and are an example of a Pythagorean

4

3

2

+ 42 = 52

triplet.

Lesson 112

Multiples of 3-4-5 are also Pythagorean

783

triplets.

3-4-5

6-8-10

9-12-15

12-16-20 Before pouring a concrete construction workers build

foundation for a building, wooden forms to hold the concrete. Then a worker or building inspector can use a Pythagorean triplet to check that the forms make a right angle. First the perpendicular sides are marked at selected lengths. Measure 4 ft and mark the board.

Measure ft and mark the

3

board.

the distance between the marks is checked to be sure the three measures are a Pythagorean triplet.

Then

Measure the diagonal. The distance from mark to mark should be 5

ft

in.

the three measures are a Pythagorean triplet, the worker can be confident that the corner forms a 90° angle. If

Activity: Application

of the Pythagorean Theorem

Materials needed by each group of 2 or 3 students: •

Two

full-length,

straightedges) •

Ruler

•

Protractor

unsharpened

pencils

(or

other

784

Saxon Math 8/7

Position two pencils (or straightedges) so that they appear to form a right angle. Mark one pencil 3 inches from the vertex of the angle and the other pencil 4 inches from the vertex.

Then measure from mark between the marks

is 5

to

mark

to see

whether the distance

inches. Adjust the pencils

if

necessary.

Trace the angle formed. Then use a protractor to confirm that the angle formed by the pencils measures 90°.

marking the pencils at 6 cm and 8 cm. The distance between the marks should be 10 cm. Repeat the

Example

1

activity,

The numbers 2

2

5, 12,

and 13 are a Pythagorean

triplet

2

because

+ 12 = 13 What are the next three multiples of Pythagorean triplet? 5

Solution

.

To find the next three multiples number by 2, by 3, and by 4.

of 5-12-13,

we

this

multiply each

10-24-26 15-36-39 20-48-52

Example 2

A

roof is being built over a 24-ft-wide room. The slope of the roof is 4 in 12. Calculate the length of the rafters needed for the roof. (Include 2 ft for the rafter tail.) 12

length of rafter

Lesson 112

Solution

785

We

consider a rafter to be the hypotenuse of a right triangle. The width of the room is 24 ft, but a rafter spans only half the width of the room. So the base of the right triangle is 12 ft. The slope of the roof. 4 in 12. means that for every 12 horizontal units, the roof rises (or falls) 4 vertical units. Thus, since the base of the triangle is 12 ft. its height is 4 ft.

4ft

12

We

use the Pythagorean theorem to calculate the hypotenuse. a 2

(4

ft)

2

16

Using 12.65

a calculator feet.

We

add

ft

we

To convert

0.65

ft

about 14

Example

3

ft

this

up

T

2

= c2

2

= c2

2

= c2

ft)

ft

160

ft

M60

ft

= c

12.65

ft

« c

find that the hypotenuse

ft

+

ft

the rafter

2

ft

we

to 8 inches.

is

about

tail.

= 14.65

ft

multiply.

x If-ir1 1

round

9 + b = C2

+ 144

to inches,

0.65

?

+ (12

2 feet for

12.65

We

ft

=

7 8 -

m

'

ft

So the length of each

rafter is

8 in.

She let out all 200 ft of string and tied it to a stake. Then she walked out on the field until she was directly under the kite. 150 feet from the stake. About how high was the kite? Serena went

to a level field to fly a kite.

786

Saxon Math 8/7

Solution

We

begin by sketching the problem. The length of the kite string is the

hypotenuse of a right triangle, and the distance between Serena and the stake is one leg of the triangle. We use the Pythagorean theorem to find the remaining leg, which is the height of

150

ft

the kite. a a a

2

2

2

+ b2 = c 2

+ (150

+ 22,500

2 ft)

2

= (200

2

ft)

= 40,000

ft

a 2 = 17,500

ft

ft

2

a = Vl7,500 a - 132

Using a calculator, about 132 ft.

we

2

ft

ft

find that the height of the kite was

LESSON PRACTICE Practice set

a.

A of

was leaning against a building. The base the ladder was 5 feet from the building. How high up

12-foot ladder

the side of the building did the ladder reach? Write the answer in feet and inches rounded to the nearest inch.

b.

Figure ABCD illustrates a rectangular field 400 feet long and 300 feet wide. The path from A to C is how much shorter than the path from A to B to C?

MIXED PRACTICE Problem set

1. (110)

2. (20 28] -

Sherman deposited $3000 in an account paying 8 percent interest compounded annually. He withdrew his money and interest 3 years later. How much did he withdraw?

What

is

the square root of the

4 squared?

sum

of 3 squared and

Lesson 112

3.

Find the

median and

(a)

(b)

mode

787

of the following quiz

Class Quiz Scores Score

4. 1461

Number

of

100

2

95

7

90

6

85

6

80

3

70

3

Students

The trucker completed the 840-kilometer haul in 10 hours 30 minutes. What was the trucker's average speed in kilometers per hour?

Use 5. (72)

6. 92]

7. 92)

ratio

boxes

to solve

problems 5-7:

Barbara earned $28 for 6 hours of work. At that much would she earn for 9 hours of work?

rate,

how

Jose paid S48 for a jacket at 25 percent off of the regular price. What was the regular price of the jacket?

Troy bought a baseball card for S6 and sold it for 25 percent more than he paid for it. How much profit did he make on the sale?

8. 11101

an item marked $1.00 was reduced 50%. When the item still did not sell, the sale price was reduced 50%. What was the price of the item after the second discount?

At a yard

9. If (36)

10. 1991

11. lll2)

60%

boys

sale

of the students

were boys, what was the

ratio of

to girls?

The points

(3, 11). (-2,

-1),

and

(-2. 11) are the vertices of

Use the Pythagorean theorem length of the hypotenuse of this triangle. a right triangle.

The frame of this kite is formed by two perpendicular pieces of wood whose lengths are shown in inches.

A

loop of string connects

the four ends of the sticks. long is the string?

How

to find the

788

Saxon Math 8/7

12.

What percent

of 2.5

What

odds of having a coin land

is

2?

(77)

13. (inv. io)

14.

are the

tails

up on

4

consecu ti ve tosses of a coin?

How much

earned in 6 months on $4000 9 percent simple interest? interest is

(1WI

deposited

15.

Complete the

at

table.

Fraction

(48)

5

Decimal

Percent (b)

(a)

8

16. Divide. Write each quotient in scientific notation: (in) f

5 x

,

10

(a)

2 x 10

17.

8

u, (b) ,

4

1.2 x 10

4 x 10

Use a unit multiplier

to convert

4

8

300 kilograms

to grams.

(50)

18. Solve for

t:

d =

rt

(106)

19. nv

'

9}

Make

shows three pairs of numbers for the function y - -x. Then graph the number pairs on a coordinate plane, and draw a line through the points. a table that

20.

Find the perimeter of

do4)

rj^

e

^c

j

n

^ e figUr e

is

this figure.

a semicircle.

Dimensions are in centimeters. (Use 3.14 for

21. tl05)

7i.)

Find the surface area of this right triangular prism. Dimensions are in feet.

22.

(a)

Write the prime factorization of exponents.

(b)

Find the positive square root of

1211

1

trillion

1 trillion.

using

Lesson 112 23.


ll04)

d when C

is

62.8

k.)

James would like to mow the lawn and wash the car but has less than 60 minutes to work. Using x for the number of minutes it will take to mow and y for the number ol minutes it will take to wash the car, write an inequality for the first sentence of this problem. Then graph the inequality in the

19.

to find

first

quadrant.

Find the perimeter of the figure at right. Dimensions are in centimeters. (Use 3.14

for/r.) 8

20. (a) (io5, 113)

(b)

Find the surface area of the cube shown. Dimensions are in feet. If

the cube contains the largest

pyramid it can hold, what volume of the pyramid? 21. Find the 1951

cylinder.

volume

the

of this right circular

31-

Dimensions are in meters.

(Use 3.14 for

22.

is

n.)

H

10-

Find the measures of the following angles:

E

(a)

ZACB

B (b)

ZCAB

(c)

ZCDE

Lesson 114

23. (70)

An aquarium

805

wide, and 20 cm deep is filled with water. Find the volume of the water in the aquarium.

24. Solve:

that

0.8m -

1.2

is

40

cm

long, 10

cm

= 6

(93)

25. Solve this inequality

and graph

its

solution:

(93)

3(x - 4) < x - 8 Simplify: 2 2" 3 26. 4 •

•

2" 1

27. 1 kilogram

f

28. (1.2) (43)

^

34J

30. (-3H-2) (85)

- 50 grams

(32)

(57)

-

^4-

29.

2

f43j

(2)(-3)

-

(-8)

3 2A

4

„

.

1.5

1 - ^ 6

+ (-2X-3) + |-5

806

Saxon Math 8/7

LESSON

115

Volume, Capacity, and Mass in the Metric System

WARM-UP Facts Practice: +

—

x

-r

Algebraic Terms (Test V)

Mental Math: a.

10110 (base

c.

10" 2

2)

_ JLJ2 ~ g 1.2 | of $1200

b.

CLIV

d.

(4

1.44

e.

i.

f.

h.

A nickel is how many cents

less

x 10 8 )

-r

Convert 250

(4

x 10

cm

$1200 reduced

to

8 )

m.

\

than 3 dimes and

3 quarters?

Problem Solving: Three tennis balls

What

just

fraction of the

by the tennis

fit

into a cylindrical container.

volume of the container

is

occupied

balls?

B NEW CONCEPT Units of volume, capacity, and mass are closely related in th( metric system. The relationships between these units are basec on the physical characteristics of water under certain standarc conditions.

We state two commonly used relationships.

One

has a volume of 1 cubic centimeter and a mass of 1 gram.

One

cubic

1 milliliter

milliliter of water

can contain of water, which has a mass centimeter

of 1 gram.

One

of water has a volume of 1000 cubic centimeters and a mass of 1 kilogram. liter

One thousand cubic which has a mass of

Example

1

centimeters can contain 1 kilogram.

Lesson 115

807

1 liter of

water,

Ray has a fish aquarium that is 50 cm long and 20 cm wide. If the aquarium is filled (a)

with water

how many

to a

depth of 30 cm,

liters of

water would be

30 cm

50 cm

20 cm

in the aquarium? (b)

Solution

what would be the mass of the water

First

we

find the

volume of the water

(50 cm)(20 cm)(30

in the

aquarium?

in the aquarium.

cm) = 30,000

cm 3

(a)

Each cubic centimeter of water thousand milliliters is 30 liters.

(b)

Each liter of water has a mass of 1 kilogram, so the mass of the water in the aquarium is 30 kilograms. (Since a 1-kilogram mass weighs about 2.2 pounds on Earth, the water in the aquarium weighs about

is

1

milliliter.

Thirty

66 pounds.)

Example 2

Malaika wanted to find the volume of a vase. She filled a 1-liter beaker with water and then used all but 240 milliliters to fill the vase, (a) What is the volume of the vase?

the mass of the vase is 640 grams, what the vase filled with water?

(b) If

Solution

(a)

vase

beaker is

the

mass of

beaker contains 1000 mL of water. Since Malaika used 760 mL, (1000 mL - 240 mL), the volume 3 of the inside of the vase is 760 cm

The

1 -liter

.

(b)

The mass (640

g) is

of the water (760 g) plus the

1400

g.

mass of the vase

808

Saxon Math 8/7

LESSON PRACTICE Practice set

a.

What

is

the

mass of

2 liters of

b.

What

is

the

volume

of 3 liters of water?

c.

When by

d.

the bottle

kilogram.

1

A tank that is hold

was

how many

cm

with water, the mass increase! milliliters of water were added

filled

How many 25

water?

long, 10

liters

cm wide, and

8

cm deep

cai

of water?

MIXED PRACTICE Problem set

1. lno>

2.

How much $7000

at 8

With two

interest is earned in 9

months on

a deposit

o:

percent simple interest?

tosses of a coin,

(Inv. 10)

3. 1551

4. 1461

(a)

what

is

the probability of getting

(b)

what

is

the chance of getting

(c)

what

are the

two heads?

two

tails?

odds of getting heads, then

tails?

On

the first 4 days of their trip, the Schmidts averaged 410 miles per day. On the fifth day they traveled 600 miles. How many miles per day did they average for the first 5 days of their trip?

The 18-ounce container costs $2.16. The 1-quart container costs $3.36. The smaller container costs how much more per ounce than the larger container?

Use 5. 1721

ratio

boxes to solve problems

how

and

6.

minutes on her typing test. At long would it take her to type an 800-word

Eve typed 160 words in that rate,

5

5

essay? 6. (65)

The ratio of guinea pigs to rats running the maze was 7 to 5. Of the 120 guinea pigs and rats running the maze, how

many were guinea pigs? 7. (74}

Kelly was thinking of a certain number. was 48, what was | of the number?

If

f

of the

number

o

8.

m

A used at a

$1500 and sold the car markup. If the purchaser paid a sales tax of 8%,

car dealer bought a car for

40%

what was the

total price of the car

including tax?

809

Lesson 115

What

(110)

the sale price of an $80 skateboard after successive discounts of 25% and 20%?

10.

The points

9.

(99}

11. !115)

12.

is

(-3, 4), (5, -2),

and

(-3, -2) are the vertices of

a triangle. (a)

Find the area of the

(b)

Find the perimeter of the

A

triangle.

aquarium

triangle.

with

the dimensions shown has a mass of 5 kg when empty. What is the mass of the aquarium when it is half full of water? glass

Complete the

table.

Fraction

(48)

20 cm

20 cm 25

Decimal 0.875

(a)

13.

Compare: a

O

b

-f

~ b

a

14. Simplify '

(a)

(6.4

a

and express each answer in

U1)

x 10

6

)(8

Percent (b)

positive and b negative if

(79)

(83

cm

is

is

scientific notation:

10" 8

x

)

6.4 x 10* (b)

8 x 10" 15.

to convert 36 inches to centimeters.

Use a unit multiplier

(50)

16.

A

= \bh

(a)

Solve for

(b)

Use the formula h is 6.

b:

(108)

17. dnv.9.107)

Find

11041

pairs

= \bh to find b

of

numbers

when A

that

is

24 and

satisfy

the

mnct i on y = -2x. Then graph the number pairs on a coordinate plane, and draw a line through the points to show other number pairs that satisfy the mnction. What is

18.

three

A

the slope of the graphed line?

Find the area Dimensions are

of in

this

figure.

millimeters.

Corners that look square are square. (Use 3.14 for k.) 6

810

Saxon Math 8/7

Find the surface area of the cube.

19. (a) (95, 105)

(b)

Find the volume of the cube.

(c)

How many

meters long edge of the cube?

is

100 cm

each

100 cm

Find the volume of the right circular cylinder. Dimensions

20. (a)

(b)

30

30

within the cylinder is the largest sphere it can contain,

1

If

the

is

volume

of

\

T

are in inches.

what

cm

100

/ Leave n as

n.

the

sphere? 21.

Find the measures of the following angles:

(40)

ZYXZ

(a)

(b)

ZWXV

ZWVX

(c)

problem 21, ZX is 21 cm, YX is 12 cm, 14 cm. Write a proportion to find WV.

22. In the figure in (97)

23. lll3)

and

A

XV is

pyramid

is

cut out of a plastic

cube with dimensions as shown. What is the volume of the pyramid?

6

6

in

Solve: 24. 0.4n + 5.2

25.^

= 12

(93)

(98)

=

y

^ 28

Simplify: 27. 3

yd -

2

ft

1 in.

(56)

28. 3.5 (43)

If

-r

V

MHM (85)

(-6)(2)

2 29. 3.5 + 2"

3

-r

5

(57)

+

(

_ 8) +

(_4)( +5 )

_

(2)(-3)

-

2" 3

in.

Lesson 116

811

LESSON

116

Factoring Algebraic Expressions

WARM-UP Facts Practice: Multiplying and Dividing in Scientific Notation (Test

W)

Mental Math: a.

101010 (base 2

+ 2~

c.

(-2)

e.

3x +

1.2

g.

125%

of

i.

Estimate

2)

2

= 2.4

$400 3a

8|%

b.

DCCCXII

d.

(5

f.

Convert

h.

sales tax

on

x 10 5 ) 1

m

x 10 2 )

(2

-r

2

to

$400 increased

cm 2 25%

,

a $41.19 purchase.

Problem Solving: Here are the front, top, and side views of an object. Draw a three-dimensional view of the object from the perspective of the upper right front.

Front

Top

Right Side

NEW CONCEPT Algebraic expressions are classified as either monomials or polynomials. Monomials are single-term expressions such as the following three examples:

6x 2 y 3

5xy

-6

~2w

Polynomials are composed of two or more terms. All of the following algebraic expressions are polynomials:

3x 2 y + 6xy'

x 2 + 2x +

1

3a + 4b + 5c + d

by the number of terms they contain. For example, expressions with two terms are called binomials, and expressions with three terms are called 2 2 2 trinomials. So 3x y + 6xy is a binomial, and x + 2x + 1

Polynomials

is

may be

a trinomial.

further classified

812

Saxon Math 8/7

Recall that to factor a monomial, we express the numerics part of the term as a product of prime factors, and we expres part of the term as a product of factor 2 3 (instead of using exponents). Here we factor 6x y

the literal

(letter)

:

6x 2 y 3

original form

(2)(3)xxyyy

factored form

Some polynomials can polynomial we first find

be

also

To

factored.

the greatest

common

factor

factor of

th


Decimal

Percent

175%

(b)

(d)

(c)

12

1801

of the

Triangle ABC with vertices A (0, 3), B (0, 0), and C (4, 0) is rotated 180° about the origin to AA'B'C. What are the coordinates of the vertices of AA'B'C?

What

is

the measure of each exterior angle and each

interior angle of a regular 20-gon?

Rob bought a jacket for $42. How much money did Rob save by buying the jacket on sale instead At a 30%-off

sale

of paying the regular price?

8. 11151

The

with interior (a)

9. (50)

Use

maximum of how many liters?

The aquarium has capacity

(b)

aquarium dimensions as shown.

figure illustrates an

30 cm

a

20

cm

40 cm

with water, what would be the mass of the water in the aquarium? If

the aquarium

is filled

a unit multiplier to convert 24 kg to lb. (Use the

approximation

10. Write

1

kg - 2.2

an equation

lb.)

to solve this

problem:

(101)

Six less than twice what

number

is

48?

815

Lesson 116 11. 101)

Find the measure of the angle of the triangle shown.

F =

12. Solve for C:

largest

1.8C + 32

(106)

13. (104)

The inside

surface of this archway

be covered with a strip of wallpaper. How long must the strip of wallpaper be in order to reach from the floor on one side of the will

archway around

20

in:

66

in.

on the other side of the archway? Use 3.14 for k and round up to the nearest to the floor

inch.

14.

What

11051

below?

is

the total surface area of the right triangular prism

15

cm 20 cm

20 cm

15. (95)

16. )A

formula to find the area of the trapezoid shown

this

above. 18.

Find two solutions

for

3x 2 -

= 40.

5

(109)

each quotient in scientific notation: A 4 8 4 x 10 8 x 10" (b) 7T 4 8 4 x 10 8 x 10"

19. Express (111)

ft

.

,

(a)

20.

m

What

is

the product of the two quotients in problem 19?

Why?

21. Factor each algebraic expression: (116)

22. {113)

f

o

-

(a)

9^7

(b)

10a 2 /) + 15a 2 b 2 + 20abc

A

playground ball just fits inside a cylinder with an interior diameter of 12 in.

What

is

volume of the n and round the

the

Use 3.14 for answer to the nearest cubic inch.

ball?

23. (a) In the figure,

what

is

mZBCD?

(40)

mZBAC? mZACD?

(b)

In the figure,

what

is

(c)

In the figure,

what

is

(d)

What can you conclude about

1 -

B

the three triangles in

the figure? 24. Refer (97)

to

the

figure

in

problem 23

proportion:

BP BC

?

CA

to

complete

this

Lesson 116

Solve: 25.

x - 15 = x + 2x +

26. 0.12(2n

1

(102)

-

5)

= 0.96

(102)

Simplify: 2

27. a(b

-

c]

+ b(c -

2g>

a)

das)

(96)

29.

(a)

(-3)

(b)

^8

2

(8x y)(12x

(4xy)(6y

+ (-2X-3) - [-2Y

(103,105)

30. If 7,35j

0.

AB

+ ^8

is 1.2

units long

75 u nit long, what

of AD?

is

and

is

the length

3

y

2 )

2 )

81

818

Saxon Math 8/7

LESSON

117

Slope-Intercept Form of Linear Equations

WARM-UP Facts Practice: +

x

Algebraic Terms (Test V)

Mental Math: a.

1000000 (base

2)

(-9H-4) c. e.

8i.

-6 ,2 = 50 2a^

12|% of $4000 Find 10% of 60, + f~,

-

b.

MCDXCII

d.

(7

f.

Convert 100°C to Fahrenheit.

h. 4,

x

8,

+

1,

x 10~ 4 )

4-

(2 x

10~ 6 )

12f % less than $4000 f, x 3, + 1, t 4, x 5,+

1,

7.

Problem Solving: In this 4-by-4 square

we

see sixteen 1-by-l

nine 2-by-2 squares, four 3-by-3 squares, and one 4-by-4 square. How many squares of any size are in this 6-by-6 square? squares,

NEW CONCEPT The

three equations

below

are equivalent equations. Each

equation has the same graph.

(a)

2x + y - 4 =

(b)

2x + y = 4

(c)

y = -2x +

4

Equation (c) is in a special form called slope-intercept form. When an equation is in slope-intercept form, the coefficient of x is the slope of the graph of the equation, and the constant

Lesson 117

819

the y-intercept (where the graph of the equation intercepts the y-axis). is

slope

y =

©

O

x

t

y-intercept

Notice the order of the terms in this equation. The equation is solved for y, and y is to the left of the equal sign. To the right of the equal sign is the x-term and then the constant term. The model for slope-intercept form is written this way:

Form

Slope-Intercept

y = mx

In this model,

Example

1

Transform

+ b

m stands for the slope and b for the y-intercept.

this

equation so that

it is

3x + y = Solution

We

solve the equation for

y by

in slope-intercept form.

6

subtracting 3x from both sides

of the equation.

3x + v 3X

equation

6

+ v _ 3 X = 6 - 3x

y =

6

- 3x

subtracted 3x from both sides simplified

Next, using the commutative property, we rearrange the terms on the right side of the equal sign so that the x-term precedes the constant term.

y =

y

6

- 3x

= -3x + 6

equation

commutative property

820

Saxon Math 8/7

Example 2 Solution

Graph y = -3x + 6 using the slope and y-intercept.

The slope

the coefficient of x, which is -3, and the y-intercept is +6, which is located at +6 on the y-axis. From this point we move to the right 1 unit and down 3 units because the slope is -3. This gives us another point on the line. Continuing this pattern, we identify a series of points of the graph

through which

is

we draw

the graph of the equation. y

Example

3

Solution

Using only slope and y-intercept, graph y = x -

The slope is

-2.

is

2.

which is +1. The y-intercept -2 on the y-axis and sketch a line that has a

the coefficient of x,

We begin at

slope of +1.

Practice set

Write each equation below in slope-intercept form: a.

2x + y =

3

c.

2x + y -

3

=

b.

y -

3

d.

x +

y= 4-

= x

x

821

Lesson 117

Using only slope and y-intercept, graph each of these equations: e.

g.

y = x -

3

f.

y = \x -

h.

2

y = -2x y = -x +

+ 6 3

dlXED PRACTICE Problem set

1. (no)

2.

How much interest is

earned in four years on a deposit of it is allowed to accumulate in an account interest compounded annually?

$20,000 if paying 7%

In 240 at-bats Chester has 60 hits.

(Inv. 10)

(a)

What

is

the statistical probability that Chester will get

a hit in his next at-bat? (b)

What

are the

odds of Chester getting a

hit in his next

at-bat?

3. l55)

4.

On On

her first four tests Monica's average score was her next six tests Monica's average score was What was Monica's average score on all ten tests?

Complete the

table.

Fraction

75% 85%

Percent

Decimal

(48)

1.4

(a) 11 12

5. 1801

of

,

m

(d)

(c)

AABC reflected

in the y-axis is AA'B'C If the coordinates of vertices A, B, and C are (-1, 3), (-3, 0), and (0, -2), respectively, then what are the coordinates of

The image

vertices A' B',

6.

(b)

The

and

'.

C7

figure at right

shows regular

ABCDEFGH. What is the measure

H

octagon (a)

B

of each

exterior angle? (b)

What

is

the measure of each

interior angle? (c)

diagonals can be drawn from vertex A?

How many

D

822

Saxon Math 8/7

7. 92)

In one year the population in the county surged from 1.2 million to 1.5 million. This was an increase of what

percent?

8. 15)

A beaker is filled with water to 500 mL level. (a)

What

the

is

volume

the

of the water

in cubic centimeters? (b)

What

the

is

mass of the water

in kilograms?

9.

Use two unit multipliers

to convert

540

2

ft

to

yd 2

.

(88)

10. Write

an equation

to solve this

problem:

(101)

Six more than three times what is 81?

11.

Find the measure

(101)

marked y in the

12. Solve for c

2 :

c

2

angle

of the

figure

number squared

shown.

- a 2 = b2

(106)

13. v 10) '

The

face of this spinner

into four sectors. Sectors

90° sectors, and sector sector. If the (a)

what

arrow is

is

the

is

divided

B and D are C is a 120°

spun once, probability

(expressed as a decimal) that will stop in sector B?

14. (112)

(b)

what

is

(c)

what

are the

the chance that

odds that

it

it

it

will stop in sector

will stop in sector

The coordinates of the vertices (3,0), (-1,-3), and (-4,1).

C?

A?

of a square are

(a)

What

is

the length of each side of the square?

(b)

What

is

the perimeter of the square?

(c)

What

is

the area of the square?

(0, 4),

823

Lesson 117

15. lll3)

A

cylinder and a cone have an equal height and an equal diameter as shown. (a)

right

What

circular

volume

the

is

the

of

cylinder? (b)

What

is

the

volume

of the cone?

Leave n as

16.

The formula

for the

volume

n.

of a rectangular prism

is

(106)

V (a)

Transform

(b)

Find h when Vis 6000

17. Refer to the

this

= lwh

formula

to solve for h.

cm 3

1 is

,

20 cm, and

graph shown below to answer

w is

30 cm.

(a)-(c).

(117)

i cn n •j

4,

f2 -1

6-5

- *-3

-1

/

—

1

2

3

4

5

6

i

± (a)

What

(b)

At what point does the

(c)

What

is

is

the slope of the line? line intersect the y-axis?

the equation of the line in slope-intercept

form?

each equation in slope-intercept form: (b) 2x + y = 4 y + 5 = x

18. Write (117)

(a)

19. Factor (116)

(a)

20.

m

each algebraic expression:

24xy 2

(b)

3x 2 + 6xy - 9x

Find the area of a square with sides Express the area (a)

in scientific notation.

(b)

as a standard numeral.

5 x 10

3

mm

long.

824

Saxon Math 8/7

21.

m

Use two unit multipliers

(97>

is

ABC is

Which side Which side to side

of ACBD corresponds

BC of AABC?

to side (b)

and CAD and CBD.

a right triangle

similar to triangles

(a)

of ACAD corresponds

AC of AABC?

23. Refer to the figure

below

to find the length of

30)

5

.

in.-

B

24. Solve this inequality

and graph

its

solution:

(93)

x + 12 < 15

6w - 3w

25. Solve:

+ 18 = 9{w -

4)

(102)

Simplify:

-

26. 3x(x

2y)

+ 2xy(x +

3)

(96)

2 1 3 27. 2" + 4" + ^127 + (-1) (57, 103, 105)

28. (-3)

+ (-2)[(-3)(-2) -

(+4)]

-

(-3)(-4)

(85)

1.2 x

10

-6

29. (ni)

to problem

20(b) to square meters.

22. Triangle

(7,

answer

to convert the

4 x 10^

2

3Q ao3)

3

36a b c 2

I2ab c

segment BD.

825

Lesson 118

E S S

O N

118

Copying Angles and Triangles

WARM-UP Facts Practice: Multiplying and Dividing in Scientific Notation (Test

W)

Mental Math: a.

101011 (base 2

c.

(-3)

e.

_ 200 A. -

g-

150%

i.

33

2)

+ 3" 2

b.

MDCCLXXVI

d.

(5

f.

300

of $4000

h.

At an average speed of 30 mph, 40 miles?

x 10

_6

x 10

)(3

Convert 7500 g

2 )

to kg.

$4000 increased 150%

how

long will

it

take to drive

Problem Solving: Sylvia wants to pack a 9-by-14-in. rectangular picture frame that is |-in. thick into a rectangular box.

^1

12

in.

9

in.

9 n j

The box has

inside dimensions of

10

12-by-9-by-10 in. Describe why you think the frame will or will not fit into the box.

14

in.

in.

NEW CONCEPT Recall from Investigations 2 and 8 that we used a compass and straightedge to construct circles, regular polygons, angle bisectors, and perpendicular bisectors of segments. We may also use a compass and straightedge to copy figures. In this lesson we will practice copying angles and triangles.

Suppose we

are given this angle to copy:

We begin by

drawing a ray •-

to

form one side of the angle. »-

826

Saxon Math 8/7

Now we second

need

ray.

to find a point

We

through which

find this point in

compass and draw an

two

to

steps. First

draw

we

the

set the

arc across both rays of the original

angle from the vertex of the angle. Without resetting the compass, we then draw an arc of the same size from the endpoint of the ray, as we show here.

Original Angle

For the second step, we reset the compass to equal the distance from A to B on the original angle. To verify the correct setting, we swing a small arc through point B while

on point A. With the compass at this setting, we move the pivot point to point A' of the copy and draw an arc that intersects the first arc we drew on the copy.

the pivot point

is

A' Copy

Original Angle

As

a final step,

we draw

through the point

at

the second ray of the copied angle

which the

arcs intersect.

Copy

We

use a similar method to copy a triangle. Suppose asked to copy AXYZ.

We

we

are

by drawing a segment equal in length to segment XY. We do this by setting the compass so that the will begin

Lesson 118

827

X

pivot point is on and the drawing point is on Y. We verify the setting by drawing a small arc through point Y. To copy

we

sketch a ray with endpoint X'. Then we locate Y' by swinging an arc with the preset compass from point X\

this segment,

first

z

Copy

Original Triangle

on the copy, we will need to draw two different arcs, one from point X' and one from point Y' We set the compass on the original triangle so that the distance between its points equals XZ. With the compass at this setting, we draw an arc from X' on the copy.

To

locate Z'

.

Copy

Original Triangle

Now we change the original.

With

this

compass we draw an

setting of the setting

to equal

arc

YZ on the

from Y' that

intersects the other arc.

Original Triangle

Copy

the arcs intersect, which we have labeled Z', corresponds to point Z on the original triangle. To complete

The point where the copy,

we draw segments X'Z' and

Y'Z'.

828

Saxon Math 8/7

Activity:

Copying Angles and Triangles

Materials needed: •

Compasses

•

Straightedges

For this activity work with a partner. Have one student draw an angle that the partner copies. Then switch roles. After each partner has drawn and copied an angle, repeat the process with triangles.

LESSON PRACTICE Practice set

a.

Use a protractor to draw an 80° angle. Then use a compass and straightedge to copy the angle.

b.

With a protractor, draw a triangle with angles of 30°, 60°, and 90°. Then use a compass and straightedge to copy the triangle.

MIXED PRACTICE Problem set

1.

m

The median home $180,000

to

price in the county increased from

$189,000 in one year. This was an increase of

what percent? 2.

To indirectly measure the height of a power pole, Teddy compared the lengths of the shadows of a vertical meterstick and of the power pole. When the shadow of the meterstick was 40 centimeters long, the shadow of the power pole was 6 meters long. About how tall was the power pole?

3.

Armando is marking off a grass field for a soccer game. He has a long tape measure and chalk for lining the field. Armando wants to be sure that the corners of the field are

m

11121

right angles.

that

4. 1501

How

can he use the tape measure

to

ensure

he makes right angles?

Convert 15 meters to feet using the approximation ~ 3.28 ft. Round the answer to the nearest foot. 1

m

829

Lesson 118

5. 98)

The

below shows one room of a scale drawing of a house. One inch on the drawing represents a distance of 10 feet. Use a ruler to help calculate the actual area of illustration

the room.

6. If a

pair of dice

is

tossed once,

(Inv. 10)

7. (99)

8.

(a)

what

is

the probability of rolling a total of 9?

(b)

what

is

the chance of rolling a total of 10?

(c)

what

are the

odds of rolling a

Use the Pythagorean theorem (4, 6) to

total of

11?

to find the distance

from

(-1,-6).

A two-liter bottle

filled

with water

(115)

9.

how many

(a)

contains

(b)

has a mass of how

cubic centimeters of water?

many

kilograms?

Write an equation to solve this problem:

(101)

Two

thirds less than half of

what number

is

five

sixths?

(102)

m

and n are parallel. If the sum of the measures of angles a and e is 200°, what is the measure of Zg.?

10. In this figure lines

m

830

Saxon Math 8/7

11. (117)

12. 11011

13. {113)

Transform the equation 3x + y = 6 into slope-intercept form. Then graph the equation on a coordinate plane.

Find the measure of the smallest angle of the triangle shown.

A cube,

12 inches on edge,

is

topped

with a pyramid so that the total height of the cube and pyramid is 20 inches. What is the total volume of the figure?

14.

The length

(101>

length of segment

12 and

c,

BD is 12. The BA is c. Using

of segment

(97)

AD.

The three triangles in the figure shown are similar. The sum of x and y is 25. Use proportions to find x and y.

16. Lina cut a grapefruit in half. (The flat surface (105)

a

write an expression that

indicates the length of segment

15.

d

b

formed

is

called a cross section.)

knew

Lina

that the surface area of a sphere

is

four times

She estimated that the diameter of the grapefruit was 8 cm, and she used 3 in place of n. Using Lina's numbers, estimate the area of the whole grapefruit peel.

the

greatest

17. Write (117}

form.

cross-sectional

the equation

Then graph

area

y - 2x +

the equation.

5

=

of the

1

in

sphere.

slope-intercept

Lesson 118

graph shown below

18. Refer to the

answer

to

831

(a)-(c).

(117)

6 5

-4 3

-2 -1

^2

6-5-4-3-2-1

3

4!

5

6

•

:

19. 06. 118)

20. (104)

(a)

What

is

the slope of the line?

(b)

What

is

the y-intercept of the line?

(c)

What

is

the equation of the line in slope-intercept form?

Draw an

estimate of a 60° angle, and check your estimate a protractor. Then set the protractor aside, and use a compass and straightedge to copy the angle.

w^ A

semicircle with a 7-inch diameter

rectangular half sheet of paper. the resulting shape? (Use

^ 7

4^

A

is

a

the perimeter of

for n.)

in.

4^

in.

11

21.

What

was cut from

in.

in.

-3

m

A

kilometer is 3 1 x 10 m. About how many dimes would be needed to make a stack of dimes one kilometer high? Express the

dime

answer in 22. Factor

is

about

1

x 10

scientific notation.

each algebraic expression:

(116)

(a)

x2 + x

(b)

2 2 12m 2 n 3 + 18mn 2 - 24m n

thick.

832

Saxon Math 8/7 Solve: 23.

-2^w 3

(93)

24.

3

\2

25,

1

1- = 4

5x z +

1

66?%

of

= 81

(109)

,-2

26.

f

|

of 0.144

(48)

27. [-3 + (-4)(-5)]

- [-4 - (-5H-2)]

(91)

Simplify: 2

2

(5x yz)(6xy z) 28. (103)

30. 11001

10 xyz

The length

29. x(x + 2) + 2(x + 2)

of the hypotenuse of

between which two consecutive whole numbers of

10

this right triangle is

millimeters?

20

mm

mm

Lesson 119

833

LESSON

119

Division by Zero

WARM-UP Facts Practice: +

—x

-f

Algebraic Terms (Test V)

Mental Math: a.

11011 (base

2)

2

c.

(2' )(-2)

+ 12 = 12

e.

a

O' i.

66|% of S600 What fraction of an hour

b.

MLXVI

d.

(i

f

Convert 5

.

x io~

8 )

t

cm 2

(i

to

x io

mm

-41

2

S600 reduced 33§% minutes less than | of an hour? h.

is

5

Problem Solving: Figure ABC is an equilateral triangle whose perimeter is 6 cm. Segment AD bisects segment BC to form two congruent right triangles. Find the length of segment

AD. and leave the answer

in irrational form.

C

NEW CONCEPT When

performing algebraic operations, it is necessary to guard against dividing by zero. For example, the following expression reduces to 2 only if x is not zero:

—x What

is

= 2

if

x *

the value of this expression

2x —

-

5

if

x is

zero?

expression

substituted

multiplied 2

for

•

x

834

Saxon Math 8/7

the value of ^? How many zeros are in zero? Is the quotient 0? Is the quotient 1? Is the quotient some other number? Try the division with a calculator. What answer

What

is

does

the

Notice

display?

calculator

that

the

calculator

displays an error message when division by zero is entered. The display is frozen and other calculations cannot be performed until the erroneous entry is cleared. In this lesson we will consider why division by zero is not possible.

Consider what happens to a quotient when a number is divided by numbers closer and closer to zero. As we know, zero lies on the number line between -1 and 1. Zero is also between -0.1 and 0.1, and between -0.01 and 0.01. -0.01^^0.01 -0.1

-1

Q

0.1

In the following example, notice the quotients

we Example

1

number by numbers

closer

and closer

get

when

to zero.

Find each set of quotients. As the divisors become closer to zero, do the quotients become closer to zero or farther from zero? (a)

Solution

divide a

we

(a)

10

10

^0_

1

0.1

0.01

10, 100,

1000

(b)

10 _10_ _J0_ -1 -0.1 -0.01

(b)

-10, -100, -1000

As

the divisors become closer to zero, the quotients farther from zero.

become

approach zero from the positive side, the quotients become greater and greater toward positive infinity (+°°). However, as the divisors approach zero from the negative side, the quotients become less and less toward negative infinity (—). In other words, as the divisors of a number approach zero from opposite sides of zero, the quotients do not become closer. Notice from example

Rather, the quotients

1 that as the divisors

grow

infinitely far apart.

As the

we might wonder whether

divisor

the quotient would equal positive infinity or negative infinity! Considering this growing difference in quotients as divisors approach zero from opposite sides can help us understand finally reaches zero,

why

division by zero

is

not possible.

Lesson 119

835

Another

consideration is the relationship between multiplication and division. Recall that multiplication and division are inverse operations. The numbers that form a multiplication fact may be arranged to form two division = 20, we may facts. For the multiplication fact

4x5

arrange the numbers to form these two division facts:

™

=

20 = 4

and

5

4

We

5

we

divide the product of two factors by either factor, the result is the other factor. see that

if

product factor a

product

,

= riactor ?

^7

^

and

=

c

factor-,

tactor 2

This relationship between multiplication and division breaks

down when Example

2

zero

The numbers

is

one of the

factors, as

we

2x3

in the multiplication fact

arranged to form two division

^ =

2.

= 6 can be

facts.

^ =

and

2

see in example

3

2

3

we

attempt to form two division facts for the multiplication = 0, one of the arrangements is not a fact. Which fact arrangement is not a fact? If

2x0

Solution

The product

and the

is

factors are 2

and

0.

So the possible

arrangements are these:

- =

— =

and

2

2

not a fact

fact

= 2

The arrangement The multiplication fact any more than 3x0 =

is

2x0

not a

fact.

0^0

=

implies

= 2 does not imply = 3. This breakdown -r

in the inverse relationship between multiplication and division when zero is one of the factors is another indication that

division

Example

3

by zero

is

not possible.

asked to graph the following equation, what number could we not use in place of x when generating a table of ordered pairs? If

we were

12

y

3

+ x

836

Saxon Math 8/7

Solution

This equation involves division. Since division by zero is not possible, we need to guard against the divisor, 3 + x, being zero. When x is 0, the expression 3 + x equals 3. So we may use in place of x. However, when x is -3, the expression 3 + x equals zero. 12

y yv =

y =

equation

—

12 -,

3

+

12 —

r

(-3)

replaced x with -3 F

not permitted

Therefore, we may not use -3 in place of We can write our answer this way:

x

in this equation.

x * -3

LESSON PRACTICE Practice set

a.

Use

a calculator to divide several different

numbers

of

your choosing by zero. Remember to clear the calculator before entering a new problem. What answers are displayed?

b.

The numbers

in the multiplication fact 7 x 8 = 56 can

be arranged to form two division facts. If we attempt to form two division facts for the multiplication fact = 0, one of the arrangements is not a fact. Which arrangement is not a fact and why?

7x0

For the following expressions, find the number or numbers that may not be used in place of the variable.

837

Lesson 119

IIXED PRACTICE

Problem set dnv.

1. w)

Robert was asked to select and hold three cards from a norma } d ec k of cards. If the first two cards selected were aces, what is the chance that the third card he selects will be one of the two remaining aces?

Khalid saved $5 by purchasing an item at a sale price of $15, then the regular price was reduced by what percent?

2. If

m 3.

(78>

On

number

graph all real numbers that are both greater than or equal to -3 and less than 2. a

the sum of the measures of the interior angles of any quadrilateral?

4.

What

5.

Complete the

m

line

is

table.

8 9

6.

(a)

Use a centimeter

ruler

triangle with legs 10

0.5%

(b)

(a)

1171

and

cm

(d)

(c)

a protractor to

draw

a right

long.

(b)

What

(c)

Measure the length of the hypotenuse

is

Percent

Decimal

Fraction

(48)

the measure of each acute angle? to the nearest

centimeter.

7.

Simplify. Write the answer in scientific notation.

(in) 5

(6

x 10 )(2 x 10 (3

8.

(a)

2x 2 + x

figure

stacking figure to 9.

)

What

at

(b)

right

2 3a 2 b - 12a + 9ab

was formed by

1-cm cubes. Refer to the answer problems 9 and 10. is

the

volume

of the figure?

(70)

10. (105)

)

4

Factor each expression:

(116)

The

x 10

6

What figure?

is

the surface area of the

7~~7

/ / /

:

838

Saxon Math 8/7

11. (108)

Transform the formula A = \bh to solve for h. Then use the transformed formula to find h when A is 1.44 m 2 and b is 1.6 m.

12. If the ratio of l65)

girls in a class is 3 to 5,

then what

percent of the students in the class are boys?

13. If {U2]

boys to

a

10-foot

ladder

leaned

is

against a wall so that the foot of the ladder is 6 feet from the base of the wall, how far up the wall will the ladder reach?

The graph below shows

perpendicular to line m. Refer to the graph to answer problems 14 and 15. line

1

6 5 /

4

m

3

-2

6-5-4-3-2-1 —

1

/

\4

5

6

i

i

14. (a)

What

is

the equation of line

What

is

the equation of line

1

in slope-intercept form?

(117)

(b)

15.

What

is

m in slope-intercept form?

the product of the slopes of line

1

and

line

m? Why?

(107)

deposited in an account paying 6% interest compounded annually, then what is the total value of the account after four years?

16. If (no)

17.

m

$8000

is

The Joneses

are planning to carpet their

home. The area

be carpeted is 1250 square feet. How many square yards of carpeting must be installed? Round the answer up to the next square yard. to

839

Lesson 119

what number may not be

18. In the following expressions, (ng)

used

for the variable?

12 (a)

12 (b)

3w

3

19. In the figure (101>

long,

and

and

c,

shown,

BA is

BD

is

x

+

units

c units long. Using

x

an ex pres sion that

write

771

B

A

indicates the length of DA. 20. In (97)

the

figure

at

right,

the

three

Find the area of the smallest triangle. Dimensions

triangles are similar.

are in inches.

21. [U3]

A

sphere with a diameter of 30 cm has a volume of how many cubic centimeters?

Use 3.14

22. 06, 118)

Draw an

estimate of a 45° angle. straightedge to copy the angle.

Then use

a

\orn.

compass and

Solve: 23. (93)

25.

-777

+ - = 4

3

— 12

x + x + 12 = 5x

24. 5(3

-

x)

= 55

(102)

26.

10x 2 = 100

(109)

(102)

Simplify: 28. x(x + 5)

27. J90,000 2

2

(12xy z)(9x y 29. cos)

- 2(x +

(96)

(20)

Z&xyz

2

z)

30

_

'"

BX

CO CD

o

3 O

c o

5

CD

CO

Notice that the value of each place is two times the value of the place to its right. To find the value of a number, we multiply the value of each digit by its place value. So 1010 (base 2) equals (1 x 8) + (1 x 2), which is ten.

1010 (base

Example

1

Solution

2)

= 10 (base 10)

What base 10 number does 10101

We

(base 2) represent?

add the values of the places occupied by 8's

16's

4's

2's

10 10

We see a

16, a 4,

and

a

1,

which

l's.

l's 1

in base 10 totals 21.

856

Saxon Math 8/7

Roman numerals

number systems use place value. The value of a Roman numeral is the same whatever its place. Here are the Not

all

values of the

Roman numerals we

will consider in this book:

I

1

V

5

X

10

L

50

C

100

D

500

M

1000

numeral, we add the values of the numerals. So MCLXII equals 1000 plus 100 plus 50 plus 10 plus 1 plus 1, which equals 1162. An exception to the rule of adding the values occurs when a numeral of lesser value is to the left of a numeral of greater value. In such a situation we subtract the lesser value from the greater value. The six possible combinations are these:

To

find the value of a

IV = 4

IX = 9

XL

= 40

XC

= 90

CD

= 400

CM

= 900

So, for example, the

Example 2

Roman

Roman numeral for 999

is

CMXCIX,

not 1M.

Carved into the base of a building was the Roman numeral

MCMXXIV,

indicating the year in which the building constructed. In what year was the building constructed?

Solution

We

will spread out the

Roman numeral

to

show

was

the value of

its parts.

Adding these

M

CM

XX

IV

1000

900

20

4

values,

we

constructed in 1924.

find

that

the

building

was

i

Topic

ESSON PRACTICE Practice set

Find the base 10 value of these base a.

Ill (base

c.

1100 (base

2)

2)

Find the value of each e.

XXXIX

g.

MCMXIX

Roman

2

numbers:

b.

1000 (base

d.

10001 (base

numeral: f.

LXIV

h.

MMII

2)

2)

A

857

Supplemental Practice

859

Supplemental Practice Problems for Selected Lessons This appendix contains additional practice problems for concepts presented in selected lessons. It is very important that no problems in the regular problem sets be omitted to make room for these problems. Saxon math is designed to produce long-term retention through repeated exposure to concepts in the problem sets. The problem sets provide enough initial exposure to concepts for most students. However, if a student continues to have difficulty with certain concepts, some of the problems in this appendix can be assigned as remedial exercises.

860

Saxon Math 8/7

Lesson 3

Find each missing number: 1.

w

4.

z

7.

6p = 48

10.

+ 36 = 62

2.

x - 24 = 42

8 = 16

5.

18 +

8.

144

-r

=

13. 8 + 6 +

14. 36

Lesson 6

For each number,

3

list

w

the

whole-number

3.

350

4.

1326

5.

4320

6.

950

7.

12,000

8.

35,420

9.

36,270

9.

24 30

3l|

36 =

4m

12.

84

factors

3600

*

9.

= 6

E_

= 90

2.

* 5

77

-

36

20

24 -

q = 8

1.

.15

6.

= 72

x + 4 = 30

+ 18 + 27 +

Reduce each

5y = 60

36

5+

from

1 to 10:

10. 123,450

11. 1,000,000

Lesson 15

—c =

11.

3

18

m

3.

12.

2520

fraction to lowest terms:

JL

9 Z *

k b>

10.

24

16 32

6g

JL

o d *

12 18

,

24

7

ft *

11.

36

8A

* 12

.

28 35

4||

Supplemental Practice

Lesson 19

Find the perimeter of each polygon. Dimensions are in inches. 1.

2. 12

20

10

15

3.

4. 12

10 14

20

12

16

18

Lesson 20

Simplify: 1.

8'

5.

3

Z

+

2'

2.

2"

6.

5

10'

io.

9.

10°

13. a/81

17.

Lesson 21

861

V900

A

52

14.

-

3

VT21

18. a[625

4'

3.

3

4.

l(r

7.

^

8.

15'

11. 5

4

-

5

3

Vl44

15. a/49

16.

19. \fl96

20. \/441

Write the prime factorization of each number: 1.

81

2.

300

3.

2000

4.

625

5.

450

6.

1200

7.

440

8.

750

9.

10,000

10.

128

780

12.

1540

11.

12. 25'

862

Saxon Math 8/7

Lesson 23

Simplify using regrouping:

4.

5§ + 3|

5.

6§ + 5§

6.

8§ + 2§

- if

u.

4 | - if

12.

6± - 2*

10. 5± ^

Lesson 26

3

5

6

6

Simplify: 1.

3§ x |

2.

2| x 3

3.

if

x

4-

7 x

5.

I 8

x 3^

6.

2^

x l

2f •i

7

-

3

h

10. 3§

Lesson 30

5

3

+ if

«-

11.

2

4

5

M

12.

f,

Simplify: 3 1.

it

10

8

3

4.

5

1

8

2

12

7.

8.

8

2

12

3

11.

3.

4

7

5.

4

10.

3

2.

t-4

9.

12.

+

+

6.

4

3 5

^h

6^3§

3|

2

i

3|

Supplemental Practice

Lesson 31

Use words 1.

16.125

2.

5.03

3.

105.105

4.

0.001

5.

160.165

6.

4000.321

Use

digits to write

each decimal number:

7.

one hundred twenty-three thousandths

8.

one hundred and twenty-three thousandths

9.

one hundred twenty and three thousandths

10. five

11.

hundredths

twenty and nine hundredths

12. twenty-nine

Lesson 33

each decimal number:

to write

Round 1.

whole number:

to

2.

two decimal

12.83333

Round 7.

to the nearest

five tenths

23.459

Round 4.

and

0.08333

3.

86.6427

6.

0.1084

9.

3.14159

places: 5.

to the nearest

164.089

6.0166

thousandth: 8.

0.45454

10.

Round 283.567

to the nearest

hundred.

11.

Round 283.567

to the nearest

hundredth.

12.

Round 126.59

to the nearest ten.

Saxon Math 8/7

Lesson 35

Simplify: 45.3 + 2.64 + 3

2.

0.4

3.

3.6 + 2.75 + 0.194 + 3

4.

12.8 + 6.32 + 15

5.

10 + 1.0 + 0.1 + 0.01

6.

278.4 + 3.26 + 1.475

7.

14.327 - 6 5

8.

10.8

9.

6.5

- 4.321

10. 10

11. 0.1

- 0.019

12. 5

- 9.67

- 4.76 - 4.937

13. 0.3 x 0.12

14. 4.5 x 5

15. 8 x 0.012

16. 0.2 x 0.3 x 0.4

17. 1.2 x 1.2 x

19. 0.144

Lesson 37

+ 0.5 + 0.6 + 0.7

1.

100

18. 1.44

20. 0.144

8

Find the area of each

triangle.

^ 12

4-

16

Dimensions are in centimeters.

1.

2.

3.

4.

5.

6.

Supplemental Practice

Find the area of each

figure.

Dimensions are in centimeters.

7

7.

865

6

8.

9_

^

10

8

6

8

10.

9.

10 15 12

18

Lesson 43

Change each decimal number mixed number:

to

reduced fraction or

a

1.

0.48

2.

3.75

3.

0.125

4.

12.6

5.

0.025

6.

1.08

Change each

fraction or

mixed number

to a

5

decimal number: 9.

7.

10.

12.

11.

6^

5-

20

6

Lesson 45

2| 5

8

Divide: 1.

0.15

3.

18

5.

12.5

7.

4.3

9.

9

11. 8

4-

0.4

4-

-f

0.5

0.04

-f-

0.01

1.8

4 0.04

2.

14.4

4. 5

0.06

4-

4 0.8

6.

288

8.

1.5

10. 4.5

12. 12.5

-r

1.2

0.12

-h

4 2.5

-H

0.5

Saxon Math 8/7

Lesson 48

Complete

this table:

Fraction 5

6

Decimal

Percent

1.

2.

1.2

3.

8%

6.

5.

8.

7.

if

0.075

9.

10.

125%

12.

11.

Lesson 49

4.

Change: 1.

40 inches

2.

200 seconds

to feet

to

and inches

minutes and seconds

Simplify: 3.

3

ft

21

in.

3

yd yd

2

ft

7 in.

1

ft

8 in.

4.

90 min

2 hr

Add: 5.

+

1

+ 6 lb

Lesson 50

Lesson 52

+ 2

5 lb 10 oz

7.

1.

24

3.

300 min

5.

500

cm

7.

100

lb to

ft

18

+ 3 gal 2 qt

1 pt

yd

2.

24

to hr

4.

300 min

m

6.

500

8.

100 pounds

2.

40 - 20 + 10 -

4.

3

_

5

oz

ft

to

cm

to s

to

mm to tons

Simplify:

4x4-4

1.

4 +

3.

5

5.

10 + 10 x 10 10

+

6x7

+ 8

s

to convert:

to in.

to

s

2 gal 3 qt 1 pt

8.

8 oz

Use unit multipliers

min 23 hr 45 min 48

5 hr

6.

+ 4

b.

2

5

+ 42 - 5 x 2 +

5x5

+

5-

5

Supplemental Practice

Evaluate: 7.

ab - be + abc

8.

xy +

—

-

if

5

if

a =

x =

abc - ab - -

if

4,

a =

b =

6,

m

-

m

if

+ xz - z

11.

12.

mn

ab - ac -

c = 2

and c =

4,

c

10.

and

and 7y = 4

8

7 9.

b =

5,

3

= | and n = | if

—

w if

=

1.2,

a =

4,

x = b =

0.5,

and z =

and c =

3,

0.1

2

c

Lesson 56

Subtract: 1.

5

2.

10

3.

4 yd 6

4.

1

8

-

-

min 13 in.

hr 10

5.

Lesson 57

7 in.

ft

1

yd yd

3

ft

10

s

-

3

-

2

ft

in.

min 28

s

8 in.

min - 24 min 40 2

ft

4

in.

2

ft

9

in.

s

6.

-

min 30 hr 48 min 43

3 hr 17

s

2

s

Simplify: 1.

4

3.

4

-2

2

2 5. 2"

2 x 4"

3 x 2"

2

4.

3

6.

2" 2 + 2" 2

8.

3(3" 3 )

7.

10" 2 x 10

9.

3 Write 10" as a decimal number.

10.

What

is

2 the reciprocal of 2~ ?

-3

2.

3

x 3

-2

867

Saxon Math 8/7

Lesson 60

Write an equation to solve each problem. Then solve the equation. 1.

What number

2.

Three

3.

What number

4.

Six tenths of 60

5.

What number

6.

Six percent of 250

7.

What number

8.

Two

9.

What number

fifths of

f of 24?

is

60

is

is

is

is

of 120?

what number?

| of 300?

is

what number?

0.5 of 50?

10.

Seven tenths of 140

11.

What number

what number?

is

75%

12. Eighty percent of

Lesson 64

what number?

30%

thirds of 90

is

what number?

0.4 of 80?

is

is

is

400

of 400?

is

what number?

Find each sum: 1.

(-36) + (+54)

3.

(-6)

2.

(-15) + (-26)

+ (-12) + (+15)

4.

(+4)

+ (-12) + (+21)

5.

(-6) + (-8) + (-7) + (-2)

6.

(-9)

+ (-15) + (+50)

7.

(+42) + (-23) + (-19)

8.

(-54) + (+76) + (-17)

9

H) K) +

-

11. (-1.7) + (-3.2) + (-1.8)

H) H) +

io -

12. (-4.3)

+ (+2.63)

869

Supplemental Practice

Lesson 66

Find the circumference of each

circle.

Dimensions are in

centimeters. 1.

2.

/

3.

>

40

f

•

7

\

•

v J 1

lea Q

1

A fnr

rr

Use

4. 5 •

Use 3.14

Leave ^ as

6.

\

28

' I

'

Use

—

\

15

•

for n.

n.

it.

•

22

Leave ^ as

for^-.

n.

Simplify: 1.

-3)

-

(-8)

2.

(-12) + (+20)

3.

+8)

-

(-15)

4.

(+6)

-

(18)

5.

-3) + (-4)

-

(-5)

6.

(+3)

-

(-4)

7.

-2)

-

-

(-4)

8.

(+2)

-

(3)

9.

-6) - (-7) +

10. (+8)

-

(+9)

-

(-12)

12. (-9)

-

(10)

-

(-11)

11.

Lesson 69

for

5. I

Lesson 68

—

22

(-3)

(8)

3) - (-1) - (-8) -

(2)

-

-

(-4)

Express each number in proper scientific notation: 2.

48 x 10" 8

4.

4 0.72 x 10"

12 0.125 x 10

6.

6 22.5 x 10"

10 17.5 x 10

8.

0.375 x 10" 8

1.

7 0.15 x 10

3.

20 x

5.

7.

10^

(+5)

Saxon Math 8/7

Lesson 75

Find each

Lesson 82

Dimensions are in centimeters.

7

1.

Lesson 77

area.

6

2.

Translate and solve: 1.

What percent

2.

Sixty

3.

Thirty

is

what percent

4.

Thirty

is

150%

5.

What percent

6.

Twenty percent

7.

What percent

8.

Twelve

75%

is

of 75 of

60?

what number?

of

is

50?

what number

of

of $5.00 of

Find the area of each

of 90?

what number?

of 40

66|%

is

is

20 •

Use 3.14

what number?

circle.

Dimensions are

for n.

Use

5.

20

—"\

*

/"

—

Leave n as

22

—-\

>

for n.

Use

—

k.

for k.

•

Use 3.14

3

in centimeters.

)

4.

(

50?

$3.50?

is

2.

/

is

6.

3 Leave n as

22

for n.

n.

871

Supplemental Practice

Lesson 83

Write each product in scientific notation: 1.

(1.2

x 10 5 )(3 x 10 6 )

2.

3.

(4.2

x 10 8 )(2.5 x 10 12 )

4.

5 7 2.5 x 10 )(4 x 10 )

5.

(4

x 10" 3 )(2 x 10" 8 )

6.

-5 7 8 x 10" )(4 x 10 -5l)

7.

(2

x 10~ 4 )(6.5 x 10" 8 )

8.

6 x 10~ 4 )(4 x 10 8 )

9.

(1.6 x

10~ 5 )(7 x 1CT 7 )

10.

7

12.

7.5 x 10

7 11. (1.4 x 10 )(8 x 10" 5 )

Lesson 93

x 10 6 )(6 x 10 3 )

x 10" 9 )(3 x 10 5 ) _8 )(4

x 10 6 )

Solve: 1.

3x -

3.

= 40

2.

15 = 2x - 19

12 + 2x = 60

4.

80 = 4x - 16

5.

8x - 16 = 56

6.

3x + 12 = 54

7.

0.8x -

8.

0.3w +

9.

~w

- 12 = 60

10.

3^/7?

-Aw

+ 20 = 8

12.

-0.2y +

5

1

= 1.4

4

11.

Lesson 101

(3

1.2

= 3

+ 30 = 120

1.4

= 3.2

Write and solve an equation for each problem:

number

1.

Six more than twice what

2.

Five less than the product of 8 and what

3.

Ten

4.

What number

5.

The sum

6.

Three fourths of what number

less

than half of what number

of

is

12

is

is

72?

number

is

is

27?

50?

more than the product

what number and 6

is

of 6

and 4?

5 less than 12?

12 less than 60?

872

Saxon Math 8/7

Lesson 102

Simplify and solve the following equations:

m

1.

5m

3.

3(x - 4) = 36

5.

What

+ 6 +

- 18 = 60

2.

3x + 20 = x + 80

4.

x + 2(x -

4)

the measure of the smallest

is

angle in this figure?

6.

Find the measure of the

largest

x - 20

angle in this triangle.

Lesson 111

Write each quotient in scientific notation: 8 x 10 1.

4 x 10

8

6 x 10 2.

4

3 x 10

3.6 x 10 3.

2 x 10

12

2.4 x 10 5.

8 x 10

7.

7 x 10

6 x 10

6.3 x 10

11.

9 x 10

4

10 10

3 x

3 x 10 6.

1 x

4

7

10

2 x 10 7

7.5 x

8

12

10

10. 5 x 10

12. 5 x 10

12

7

6 4 x 10

8

8

5 4 x 10

8.

11

6

4.

6

9

1.8 x 10

1.2 x

12

7

4.2 x 10

9.

6

3

10

= 24 - x

absolute value The distance from the graph of a number to the number on a number line. The symbol for absolute value is a vertical bar on each side of a numeral or variable, e.g., |-x|.

3

3 units

units *

,

*

,

v

-3-2-10 1

1

1

\+3\

1

1

3

2

1

= \-3\ = 3

Since the graphs of -3 and +3 are both 3 units from the number 0, the absolute value of both numbers is 3.

acute angle

An angle whose measure is between 0° and

right

acute angle

\

obtuse angle

not acute angles

acute angle

An

angle

90°.

is

smaller than both a right angle and an

obtuse angle.

acute triangle

A

between 0° and

90°.

triangle

whose

One

of

two

angle measures

not acute triangles

acute triangle

addend

largest

or

more numbers

that are

added

to

find a sum. 7 + 3 = 10

additive identity of addition.

The addends

problem are 7 and

in this

The number

0.

3.

See also identity property

7+0=7 f

additive identity

We

call zero the additive identity

number does not change

because adding zero

to

any

the number.

873

874

Saxon Math 8/7

adjacent angles

Two

common common

The angles

vertex.

angles that have a lie

on

common

side and a opposite sides of their

side.

Zl and Z2 are adjacent angles. They share a common side and a

adjacent sides

common

In a polygon,

vertex.

two sides

that intersect to form

a vertex. A

AB and BC are adjacent sides. They form vertex B. D

C

algebraic addition

numbers We

to

The combining

(-3)

Any

sum

of -3, +2,

and -11:

+ (+2) + (-11) = -12

process for solving a mathematical problem.

In the addition algorithm,

and then

and negative

form a sum.

use algebraic addition to find the

algorithm

of positive

we add

the ones

first,

then the tens,

the hundreds.

alternate exterior angles

A

when

a transversal intersects

angles

lie

special pair of angles formed

two

lines. Alternate exterior

on opposite sides of the transversal and

are outside

the two intersected lines.

Zl and Z2 are alternate exterior angles. When a

transversal

intersects parallel lines, as in this figure, alternate exterior

angles have the same measure.

875

Glossary

alternate interior angles A special pair of angles formed when a transversal intersects two lines. Alternate interior angles lie on opposite sides of the transversal and are inside

the two intersected lines.

/LI

and Z2

are alternate interior angles.

When

a transversal

intersects parallel lines, as in this figure, alternate interior

angles have the same measure.

altitude

The perpendicular distance from the base

of a

triangle to the opposite vertex; also called height.

altitude

angle

The opening

segments

that

is

formed when two

lines, rays, or

intersect.

These rays form an angle.

angle bisector A line, ray, or line segment that divides an angle into two equal halves.

VT is

an angle bisector. It divides ZRVS into two equal halves.

v

arc

s

Part of a circle. A

B

The portion of the

circle

between points

A and B is arc AB.

876

Saxon Math 8/7

area The size of the inside of a in square units. 5

flat

.

associative property of addition

does not affect a + {b + c) = [a + not associative. + 4) + 2 = 8 + is

measured

The area of J this rectangle °

in.

is

Addition

is

in.

2

(8

shape. Area

+

c.

+ 2)

(4

The grouping

of addends

sum. In symbolic form, Unlike addition, subtraction is

their b)

,

,

10 square inches.

(8

- 4) - 2 * 8 -

Subtraction

associative.

is

(4

- 2)

not associative.

property of multiplication The grouping of factors does not affect their product. In symbolic form, a x (b x c) = [a x b) x c. Unlike multiplication, division is not associative. associative

(8 x 4) x 2

= 8 x

Multiplication

is

(4 x 2)

(8

4)

-r

-r

Division

associative.

is

2 ± 8 +

(4

-r

2)

not associative.

The number found when the sum of two or more numbers is divided by the number of addends in the sum; also called mean. average

To find the average of the numbers

5, 6,

and

10, add.

5 + 6 + 10 = 21

There were three addends, so divide the

sum by 3.

21 + 3 = 7

The average of 5,

base

(1)

6,

and 10

A designated side

is 7.

(or face) of a

base (2)

in

base s

means

A base

base

The lower number

5

geometric figure.

an exponential expression.

—

5

3

exponent

5x5x5, and its value is 125.

Glossary

bisect

To divide

a

877

segment or angle into two equal halves.

Y

Line

1

bisects

20

bisects

ZAMC.

A

method of displaying data that the numbers into four groups of equal size.

box-and-whisker plot involves splitting

Ray MB

XY.

40

30

50

60

box-and-whisker plot

The point inside a circle or sphere from which points on the circle or sphere are equally distant. center

10

The center of

circle

A

is

all

cm

2 inches from every point on the

circle.

The center of sphere B

is

10 centimeters from every point on

the sphere.

central angle

An angle whose vertex is the

center of a circle.

A

ZAOC is a central angle.

of expressing the likelihood of an event; the probability of an event expressed as a percent.

chance

A way

The chance of snow is 10%. There

is

an

80% chance

It is

of rain.

not likely It is

to

snow.

likely to rain.

Saxon Math 8/7

chord

A segment whose endpoints lie on a circle. AB is a

circle

A

chord of the

which

closed, curved shape in

shape are the same distance from

its

all

circle.

points on the

center.

circle

circle

A

graph

method

of displaying data, often used to

show information about percentages or parts of a whole. A circle graph is made of a circle divided into sectors. Class Test Grades

This circle graph shows data for a class's test grades.

The perimeter

circumference

of a circle.

If the

distance from point

around

to

point

A

is

A

3 inches,

then the circumference of the circle is 3 inches.

coefficient variable(s) in

coefficient

common

In

number that multiplies If no number is specified,

use, the

an algebraic term.

the

the

is 1.

In the term -3x, the coefficient is -3. In the term

y

2 ,

the coefficient

is 1.

commutative property of addition Changing the order of addends does not change their sum. In symbolic form, a + b = b +

a.

Unlike

addition,

subtraction

is

not

commutative. 8 + 2 = 2 + 8

Addition

is

commutative.

8-2*2-8 Subtraction

is

not commutative.

l

879

Glossary

commutative property of multiplication Changing the order of factors does not change their product. In symbolic form, a x b - b x

Unlike

a.

multiplication,

division

is

not

commutative.

8x2 Multiplication

=

2x8

is

commutative.

A tool used to

compass

2*2

8 +

—

Division

draw

radius

circles

and

+ 8

not commutative.

is

arcs.

gauge

pivot point

—

marking point

two types

of

Two

complementary angles

compasses

angles

whose sum

is 90°.

A

ZA and Z.B are complementary angles.

A

complex fraction fractions in

its

fraction

that

contains one or more

numerator or denominator.

25|

15

1

100

7l

2

complex fractions

not

A counting

composite number

12 101

number

9

is divisible

1 1 is

by

divisible

compound

z

complex fractions

greater than 1 that

by a number other than itself and number has three or more factors. divisible

xy_

1.

Every composite

and 9. It is composite. and 11. It is not composite.

1, 3,

by

1

interest

Interest that pays

on previously earned

interest.

Compound $100.00 +

$6.00

$106.00 +

Interest

Simple Interest $100.00

principal first-year interest total after

is

(6%

of $100.00)

one year

$6.36

second-year interest

$112.36

total after two years

(6% of $106.00)

principal

$6.00

first-year interest

$6.00

secon d -year in terest

$1 1 2.00

total after

two years

880

Saxon Math 8/7

Two or more circles with a common center.

concentric circles

common center of four

concentric circles

Having the same

congruent

and shape.

size

These polygons are congruent They have the same size and shape.

constant

A number whose value

In the expression

while r

is

does not change.

2k r, the numbers 2 and n are constants,

a variable.

construction

Using a compass and/or a straightedge

to

create geometric figures.

These tools are used in construction.

compass

coordinate(s)

number

(1)

A number

used

to locate a point

on

a

line.

A +

-3-2-10 1

1

The coordinate of point A

1

1

1

1

1

2

3

is -2.

An

ordered pair of numbers used to locate a point in a coordinate plane. (2)

y

The coordinates of point B are {2,3). The x-coordinate is listed first,

-3

the y-coordinate second.

881

Glossary

plane A grid on which any point can be identified by an ordered pair of numbers.

coordinate

Point 3-2- 1..

-

2

1

A

is

at (-2, 2)

3

located

on

this

coordinate plane.

corresponding angles

A special

pair of angles formed

when

a transversal intersects two lines. Corresponding angles lie on the same side of the transversal and are in the same position

two intersected

relative to the

Zl and Z2 intersects

lines.

When

are corresponding angles.

parallel lines,

as in

this

a transversal

figure, corresponding

angles have the same measure.

corresponding parts that

occupy the same

Sides or angles of similar polygons relative positions.

BC corresponds to YZ. ZA corresponds to ZX.

counting numbers of the set 1,

24,

The numbers used

(1, 2, 3, 4, 5, ...}.

to count; the

Also called natural numbers.

and 108 are counting numbers.

-2, 3.14, 0,

members

and 2^ are not counting numbers.

882

Saxon Math 8/7

cross product

and

The product

of the numerator of one fraction the denominator of another. 5 x 16 = 80

20 x 4 = 80

The cross products of these two fractions are equal.

decimal fraction

A

decimal number.

36.28 and 9.12 are decimal fractions. 3j

and

are not decimal fractions.

decimal number 23.94

is

A numeral that contains a decimal point.

a decimal

number because it contains

The symbol

decimal point

in a decimal

a decimal point.

number used

reference point for place value. 34.15 t

decimal point

degree

(°)

(1)

A unit for measuring angles.

tL

(2)

There are 90 degrees

There are 360 degrees

(90°) in a right angle.

(360°) in a circle.

A unit for measuring temperature. S7\ 100°C -

Water

boils.

There are 100 degrees between the freezing

and

boiling points

of water on the Celsius scale. Water freezes.

denominator

The bottom term

of a fraction.

5

numerator

9

denominator

as a

Glossary

883

dependent events Two events are dependent if the outcome of one event affects the probability that the other event will occur. Whitney draws a card from a regular deck and does not replace it. Then Chad draws a card from the same deck. The probability of Chad's drawing the queen of hearts is dependent on what card Whitney draws. If Whitney draws the queen of hearts, the probability of Chad's drawing it is 0. If Whitney draws a different card, the probability of Chad's drawing the queen of hearts is j^.

A

diagonal

two

line segment, other than a side, that connects

vertices of a polygon.

The distance

diameter

across a circle through

The diameter of this

The

difference

12 - 8 = 4

digit

Any

its

center.

circle is 3 inches.

result of subtraction.

The difference

in this

problem

is 4.

of the symbols used to write numbers:

0, 1, 2, 3,

4, 5, 6, 7, 8, 9.

The

last digit in the

directed

numbers

number 7862 is

2.

See signed numbers.

property addends is equal to the individual addend: a x distributive

number times the sum of two sum of that same number times each

A

[b

+

c]

=

(a x b)

+

[a x

c).

8 x (2 + 3) = (8 x 2) + (8 x 3) Multiplication

dividend

is

distributive over addition.

A number that is 4

12 + 3 = 4

3)12

divided.

The dividend

12 in each of these problems. is

884

Saxon Math 8/7

Able

divisible

to

be divided by a whole number without

a

remainder. The number 20 is divisible by 4, since 20 + 4 has no remainder.

5

4j20

6R2

The number 20

3)20

divisor

(1)

is

not divisible by

A number by which another number is 4

The divisor

12

3jl2

12 + 3 = 4

= 4

is

divided. 3 in each

of these problems.

3 (2)

3,

since 20 + 3 has a remainder.

A factor of a number. 2 and 5 are divisors of 10.

double-line graph A method of displaying a set of data, often used to compare two performances over time. Compounded Value

of

$1000

at

7% and 10%

Interest

double-line graph

10

edge

12

A line segment formed where two faces of a polyhedron

intersect.

One edge of this cube color.

equation

A

A statement that uses the

two quantities x = 3

symbol "="

to

show

that

are equal. 3 + 7 = 10 equations

equilateral triangle

same

is in

cube has 12 edges.

A

triangle in

4+1

x
, 527

527 than or equal to, not equal to, ^, 7 less than, , 22, 527 greater than or equal to, >, 527 of inclusion, 9, 356-358, 432-434, 571 less than,