Mathsbank KS2 Essentials: Decimals and Percentages Workbook 9780473486457, 047348644X, 9780473486440

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Mathsbank KS2 Essentials Decimals and Percentages Workbook

By J E Allen

Copyright John Edward Allen, 2019, All Rights Reserved. No part of this publication may be reproduced, stored, or transmitted in any form or by any means, electronic or mechanical, without prior permission in writing from the author. An exemption is granted to the purchaser of the physical book to photocopy pages for multiple children or students. Written by J E Allen BA. MA. M.Ed. The author welcomes any comments or inquiries from users of this material. Please send these to: [email protected] Published by Morgan Lee Clasper Contact at: [email protected] ISBN: 978-0-473-48645-7 (Kindle) ISBN: 978-0-473-48644-0 (ePub) About the Author: John Edward Allen is an author and retired headmaster with more than 35 years of experience teaching mathematics in Primary Schools across the UK. He holds a BA in Pure Mathematics, an MA in Curriculum Studies, and an M.Ed. in Mathematical Development. He founded Mathsbank in 1996, has been selling workbooks for over 19 years, and is dedicated to providing educational Mathematics material across the globe.

Welcome to Mathsbank KS2 Essentials Part One: Decimals Decimals - Introduction Chapter 1: Decimal Grids Chapter 2: Fractions and Decimals Chapter 3: Addition and Subtraction of Decimals Chapter 4: Multiplication of Decimals Chapter 5: Division of Decimals Conclusion Part One Decimals Answers Part Two: Percentages Percentages - Introduction Chapter 6: Fractions and Percentages Chapter 7: Percentages of Whole Numbers Chapter 8: Percentages in Problems Conclusion Part Two Percentages Answers

Welcome to Mathsbank KS2 Essentials Thank you for choosing Mathsbank KS2 Essentials. Inside this workbook, you’ll learn a variety of topics covering decimals, percentages, and their relationships to fractions and whole numbers. Each area is covered using clear explanations, a wealth of examples, and includes exercises for you to test your skills and apply what you’ve learned. Basic maths skills are essential for everyone to learn, regardless of their style and type of education. This workbook covers the basics in decimals, working up to three decimal points, percentages, including using whole numbers and currency, and the relationships between decimals, percentages, and fractions. An understanding of fractions is encouraged before using this book, but not essential. This workbook was originally created for supplementary use alongside the Key Stage 2 UK National Curriculum, and so is designed to both prepare students for more complex mathematical concepts, as well as supplement any courses, books, or programs they are already using. What You’ll Learn: In Book One: Understanding decimals with columns and grids Working with decimals in the tenths, hundredths, and thousandths Changing decimals to and from fractions Working with decimals on the number line Adding and subtracting decimals up to three decimal points The multiplication of decimals, with approximation Dividing decimals with a simple three-step rule Recurring or repeating decimals 26 exercises and bonus rounds… …along with over 300 questions to test your skills

In Book Two, you’ll find out about: Working with percentages and grids The relationship between percentages and fractions on the number line Changing fractions to and from percentages The “cancelling down” technique for simplifying fractions Finding percentages of whole numbers Using percentages with currency Applying percentages knowledge to real-world problems 17 exercises and bonus rounds… …and over 260 questions to test your skills Who This Book Is For: Mathsbank KS2 Essentials is for anyone looking to improve their children’s education and proficiency in maths. This includes homeschool and unschool families, as well as parents who want to supplement their children’s public or private education with additional resources. This book can be used to help struggling students achieve mathematical proficiency, as well as to test students already confident in their abilities. How To Use This Book: This book is designed to be used primarily as a workbook. The explanations given go hand-in-hand with the exercises and questions to explain, demonstrate, and test each skill in a methodical fashion. Each exercise is brief, and clearly linked to an answers section, so students can check that they’re on the right track before moving onto the next section. For the physical workbook, ample space is provided for students to work within the book, or alternatively, you can copy the questions onto your own paper – this is necessary for the eBook version. About the Author J E Allen is a retired headmaster with 35 years of experience teaching mathematics in Primary Schools across the United Kingdom. Because of this,

he has a great deal of familiarity with the schooling system, effectively teaching students, and early mathematics. He founded Mathsbank in 1996, and prior to being republished, his wide range of workbooks have (and continue to) sell thousands of copies on other platforms since the year 2000, and have helped countless students with their education. Message from the Publisher Being homeschooled myself from an early age, my grandfather’s maths books played a vital and important part of my education – many of the areas I was struggling in became suddenly easy through his methods and explanations. I would go so far as to consider it a major turning point. So years later, when I delved into the world of copywriting and publishing, it’s with great honour that I have a chance to rebrand and republish his books to reach and help an even broader, international audience. I hope that you find this book as helpful and enlightening as I did. - Morgan Lee Clasper

Part One: Decimals

Decimals - Introduction It is important that children have a solid understanding of place value for positive whole numbers before attempting to introduce them to the decimal extension of the place value system. They also need to have a sound understanding of fractions, since the conceptual ideas underpinning decimals are the same as those underpinning fractions. A decimal number comprises a whole number and a fraction of a number. It’s a way of writing a number that is not whole. Decimal numbers are 'inbetween' numbers – for example, 7.38 is in between the numbers 7 and 8. It is more than 7, but less than 8. The numbers to the left of the decimal point are normal whole numbers. The numbers to the right of the decimal point are parts of whole numbers. To find out exactly what a decimal number represents, we use place value headings, in tenths, hundredths, thousandths etc. as in the grid below.

The number in the grid represents: 4000 + 400 + 40 + 4 +

+

+

For each place to the left the value of the digit increases by ten times. For each place to the right, the value of the digit decreases by ten times. Decimals are part of the set of numbers and can be added, subtracted, multiplied and divided in the same way as with whole numbers subject to the

following criteria. There are a few key rules to keep in mind, which will be elaborated upon later in this book. 1) When adding or subtracting decimals, you must only add like values. Lining up all the decimal points will ensure this happens. You can then add, or subtract, as normal as if the decimal point wasn't there.

Notice how the decimal point is always in the same place. 2) When multiplying decimals, carry out the operation as you would with normal multiplication, but ensure that there are an equal number of places after the decimal point in your answer. 3) When dividing decimals, increase the value of the divisor and the dividend by increments of 10 times until the divisor is a whole number, then divide as normal, keeping the decimal point in the same vertical line. In our number system, the number 2345 has the same value as 00002345. Zeros in front of a whole number do not alter the value of the number. Likewise, placing zeros after the final decimal place does not affect the value of that decimal number. For example, the numbers 5.23, 5.230 and 5.2300 all have the same value: 5.23 means 5 and 2 tenths and 3 one-hundredths and 0 thousands and so on. The last zero doesn’t actually need to be there. However, extra zeros are used sometimes as place-keepers. For example, if you are required to add 2.3 and 3.45, we would add a zero to 2.3 so that they have equal places. The calculation would be set out like this:

The numbers 5.23 and 5.023 however, do not have the same value. 5.23 means 5 and 2 tenths and 3 one-hundredths like before, but 5.023 means five and zero tenths and 2 one hundredths and 3 one-thousandths. The zero in the tenths changes the value of the number. Decimals can be converted to and from percentages, i.e.

2.5 = 2

=

=

= 250%

And fractions can be converted to decimals simply by dividing the numerator by the denominator (dividing the top number by the bottom one)

Likewise, decimals can be converted into fractions like this:

0.25 =

=

When you’re comparing two decimals to work out how to order them, consider the whole numbers first. If they’re equal, then consider the fractional part of the decimal.

Chapter 1: Decimal Grids When we write numbers, the position (or "place") of each digit is important. Look at the number below. It has been placed in a grid to show its position.

The number is 777. The 7 in the first column is 7 units, the 7 in the next column is 7 tens, and the 7 in the final column is 7 hundreds. To arrive at the number we simply add the value in each column like this: 700 + 70 + 7 = 777 Notice that the value of the digit in each column is multiplied by ten. If we add another column, this column becomes the 1000’s column.

Now we have: 7000 + 700 + 70 + 7 = 7777

If we move in the opposite direction we see that the value of each column decreases by a factor of 10 each time.

Look at the next example:

Notice that as the digit 5 moves one place to the left, it increases its value by 10 times. Likewise, if we move a digit to the right, each time we must divide it by 10.

We can increase the number of columns to the left as many times as we like to accommodate larger numbers.

Each column is 10 times larger than the next. It follows, then, that we must be able to extend this system to the right. Look at the table below. If we add another column to the right of the unit column, it has to be 10 times smaller than one unit. I.e. one tenth.

The double line on the grid above is to show the point between the units column and the tenths column. So the number 234.4 is read as ‘two hundred and thirty-four point four’ and looks like:

200 + 30 + 4 +

= 234

= 234.4

Exercise 1A: 1) Write the value of each number in the space to the right of the table.

Write down the following numbers in digit form: 2) Thirty thousand and seventy 3) One hundred and seventy-five thousand, two hundred. 4) Six hundred and forty thousand, five hundred and three. 5) Eighty-eight thousand and seventeen

6) Three hundred and two thousand, two hundred and seven 7) Eight hundred and one thousand, five hundred and nineteen 8) Fifty-five thousand and ten 9) One thousand and three 10) Two hundred and eighty-eight thousand, five hundred Check Your Answers Exercise 1B: Write the expanded form of the following decimal numbers using the following format: 200 + 30 + 4 + 1) 382.7 = 2) 475.3 = 3) 774.6 = 4) 605.8 = 5) 400.3 = 6) 678.8 = 7) 986.9 = 8) 302.6 = 9) 130.7 =

= 234

= 234.4

10) 25.8 = 11) 81.4 = 12) 50.5 = Check Your Answers Hundredths: If we extend the grid one more place to the right we get:

This reads ‘Five thousand, six hundred and forty-seven point thirty-six’ and looks like:

5000 + 600 + 40 + 7 + Exercise 1C: Expand the following: 1) 338.27 = 2) 667.56 = 3) 304.45 = 4) 800.87 = 5) 773.92 = Check Your Answers

+

= 5647.36

We know from working with numbers that any number, when it is multiplied by one, the answer remains the same. For example, 36 x 1 equals 36. This is true for any number, fraction, percentage, or decimal. It is also true that one can be represented by , , ,

So

x

=

and so on.

=

The following is also true:

+

=

+

=

Therefore the decimal 243.34 can be written as 243 Exercise 1D: Expand the following as above. 1) 576.27 = 2) 549.56 = 3) 304.45 = 4) 602.87 = 5) 329.92 = How would you write 444.09? 6) 444.09 = Expand the following decimals.

7) 408.07 = 8) 400.09 = 9) 604.05 = 10) 702.08 = 11) 629.01 = Check Your Answers Note that in all cases the zero acts as a place keeper. Now for something a little harder. In this case the format would look like: 28,535.65 = 20,000 + 8,000 + 500 + 30 + 5 +

+

Exercise 1E: Expand the following decimals in the same format as above: 1) 58,234.7 = 2) 73,400.9 = 3) 103,619.7 = 4) 10,132.4 = 5) 29,000.4 = 6) 203,034.1 = 7) 99,029.5 =

8) 23,308.27 = 9) 49,030.34 = 10) 60,290.82 = 11) 903,009.04 = Check Your Answers If we extend the grid one more place to the right we get:

This reads ‘Six thousand, nine hundred and forty-eight point three six four’ and is often referred to as a decimal to three places. Expanding this, we get:

6000 + 900 + 40 + 8 +

+

Exercise 1F: Expand the following as above: 1) 45.264 = 2) 60.235 = 3) 98.456 = 4) 48.679 = 5) 909.894 =

+

= 6948.364

6) 140.073= Check Your Answers You will notice that the value of the digit in the thousandths is very small, and although we can extend decimals to as many places as we like, we tend to work to three places of decimals only (unless we have a specific reason for not doing so).

Chapter 2: Fractions and Decimals For every fraction, there is an equivalent decimal, and vice versa. For example:

0.34 =

, 0.06 =

, and 0.001 =

Exercise 2A: Write the equivalent fractions to the following: 1) 0.27 = 2) 0.14 = 3) 0.004 = 4) 0.012 = 5) 0.104 = 6) 0.111 = Check Your Answers To change any fraction to its decimal equivalent you simply divide the numerator by the denominator (The bottom number into the top number).

We have added a decimal point after the one, and a zero after the point. This is because we can see that 2 is greater than 1, and adding zeros after the decimal point does not alter the value of one.

We now have 0.5, and this tells us that one half equals 0.5.

Exercise 2B: Try these using the same method. Some of them get a bit harder toward the end. Check chapter 3 for more detail into carrying numbers.

1)

=

2)

=

3)

=

4)

=

5)

=

6)

=

7)

=

8)

=

9)

10)

=

=

Check Your Answers A decimal is simply a way of writing a number that is not a whole number. They can be thought of as 'in-between' numbers. For example, 7.4 is in between the numbers 7 and 8. It is a bit more than 7, but not as large as 8. This is shown clearly in the number line.

Exercise 2C: Write the value indicated by the black line.

Each decimal occupies a unique spot on the number line. Below is the part of the line between 0 and 2, and shows increments of one-tenth. If the space between each increment was divided into ten even smaller parts, then the line would show the position of hundredths. This repeats with thousandths, and so on.

Write the value of the point on the number line shown by each arrow.

Write the value of the point on the number line shown by each arrow.

Check Your Answers The decimal part of any number is to the right of the decimal point, and whole numbers are found to the left of the decimal point.

The decimal part of the number above is 0.27. We write a zero in front of the decimal point to show that the point and subsequent decimal numbers (the .27) will follow. We can now express all decimal parts in terms of fractions. For example:

0.34 =

. Likewise,

= 0.71 and so on.

Exercise 2D: Express the following fractions as decimals.

1)

=

2)

=

3)

=

4)

=

5)

=

6)

=

7)

=

8)

=

9)

=

10)

=

11)

=

12)

=

Convert the following decimals into fractions: 13) 0.4 = 14) 0.9 = 15) 0.7 = 16) 0.31 = 17) 0.08 = 18) 0.07 = 19) 2.5 = 20) 7.4 = 21) 8.3 = 22) 6.18 = 23) 9.83 = 24) 26.09 =

25) 39.03 = 26) 109.07 = Check Your Answers Bonus Round: Express the following decimals as fractions: 1) 0.42 = 2) 0.96 = 3) 0.77 = 4) 0.07 = 5) 0.09 = 6) 0.04 = 7) 0.112 = 8) 0.239 = 9) 0.302 = 10) 0.231 = 11) 0.039 = 12) 0.009 = 13) 0.409 = 14) 0.759 =

Check Your Answers

Chapter 3: Addition and Subtraction of Decimals The process by which we add two decimals is identical to the process used in normal addition. As with normal addition, you must make sure the units column in each number is in the same vertical line. An example will make this clear.

Notice that if you line up the units column you also line up the decimal points, the 10’s, 100’s, and all the other columns. Exercise 3A: Complete the following decimal additions. Set each one out as above. 1) 444.5 + 321.3 = 2) 302.13 + 371.24 = 3) 271.6 + 428.3 = 4) 835.61 + 144.28 = Now for something a little harder. All of the same rules apply, except this time there are three decimal points. 5) 389.553 + 129.233 = 6) 309.136 + 931.245 =

7) 989.639 + 444.328 = 8) 892.603 + 897.981 = Check Your Answers Look at the next example. This time the decimals do not have the same number of places after the decimal point. Example: 253.4 + 342.36 = ?

To solve this, we simply line up the columns as before and assume that a zero is in the space above the 6 in the 100’s column. This is true since 234.4 is identical to 234.40 as the zero is just a place-keeper. In fact, you can put as many zeros as you like after last decimal place without affecting the value of the number. So 234.4 equals 234.40, which also equals 234.400 and 234.4000 and so on. If we carry out the following addition we must adjust the columns like this:

We first add the units and find that 2 + 9 =11. We write 1 in the units column and carry 10 forward into the next column. This increases the 10’s column by one to make 5 + 2 =7. The addition of any number of decimals works in exactly the same way. Instead of adding units, ten’s etc. we are adding tenths and hundredths.

This time we write: turn this into

+

=

, but

equals

+

, so we can

+

So we must add one more

to the 10th‘s column.

Exercise 3B: Try and solve these: 1) 325.24 + 241.37 2) 271.47 + 426.36 3) 431.58 + 457.37 4) 390.69 + 509.25 5) 716.6 + 241.7 6) 628.7 + 101.8 Check Your Answers If you are required to add two decimals with a different number of places after the decimal point you proceed like this. Add 62.5 + 104 .32

Remember, always keep the decimal point in the same line. In the first decimal simply put a zero in the column to keep the hundredths place

Exercise 3C: Now try these: 1) 325.2 + 205.37 2) 271.47 + 26.3 Set the following out as above. 3) 24.3 + 127.24 4) 76.42 + 277.2 5) 96.05 + 237.9 6) 44.03 + 247.4 7) 88.4 + 77.26 8) 65.07 + 47.8 9) 8.394 + 72.2 10) 8.497 + 77.02 11) 46.051 + 397. 4 12) 74. 03 + 147.004

13) 108.401 + 77. 06 14) 65.075 + 7.008 Check Your Answers Subtraction of decimals Subtracting decimals uses the same format as the addition of decimals. The important thing to remember is that the unit’s column of both numbers must line up. By doing this, you ensure that the decimal point will be in a straight line, and so will all the other columns. To make your calculations correct you must always subtract like quantities from like quantities. Exercise 3D: Complete the following: 1) 355.2 - 241.1 2) 755.4 - 321.3 3) 795.5 - 420.2 4) 455.34 - 241.22 5) 894.67 - 137.23 6) 185.06 – 33.04 7) 361.76 – 41.22 8) 497.72 – 64.31 9) 972.53 – 661.02

10) 859.47 - 537.26 11) 795.55 - 301.43 Check Your Answers The same method is used when there is a carrying figure involved. Study the example below.

In the example, we started the calculation with the hundredth’s column and attempted to take 6 hundredths from 5 hundredths. We found that we couldn’t, so we had to borrow 1 tenth from the tenth’s column. 1 tenth is equal to 10 hundredths, and so if we add those to the hundredths that are there already there’ll now be 15 hundredths in that column. We can now take 6 from this total making 9, and the tenth’s column has been reduced by 1. Now we can complete the subtraction normally. Exercise 3E: Complete the following. 1) 505.54 - 341.17 2) 766.42 - 447.16 3) 990.55 - 601.26 In the following, you need to borrow 1 unit from the units column, which equates to 10 tenths. 4) 605.5 - 381.7

5) 964.4 - 541.5 6) 622.5 - 601.7 7) 745.2 - 344.7 8) 572.1 - 171.5 9) 847.2 - 406.5 Check Your Answers In the next examples, you will need to borrow from both the hundredth’s column and the tenth’s column. Study the example.

We tried to take 6 from 5 in the hundredths column and had to borrow from the tenths column. We then took 6 from 15. The tenth’s column was reduced to 4 and we cannot take 6 from 4 so we must borrow 1 from the unit’s column leaving it with 5 units. But 1 unit is equal to 10 tenths and so we now have 14 tenths in the tenths column. We can now subtract 8 from 14 and the rest is straightforward. Exercise 3F: Complete the following. 1) 905.54 - 301.77 2) 469.42 - 147.58 3) 793.55 - 302.56

4) 727.54 - 304.85 5) 509.10 - 247.58 6) 673.01 - 102.37 Check Your Answers

Chapter 4: Multiplication of Decimals Take a look at these examples.

1)

x3=

, therefore 0.1 x 3 = 0.3

2)

x3=

, so 0.2 x 3 = 0.6

And 3)

x3=

, so 0.3 x 3 = 0.9

We know that decimals are part of the number system and therefore must obey the same rules as normal numbers. But what about 0.3 x 0.3? From the diagram below we see that 0.3 x 0.3 = 0.09

x

=

= 0.4 x 0.4 = 0.16

In each case, the multiplication was the same as normal multiplication, but since we have two places of decimals in the sum we must have the same number of places in the answer. There is a simple rule for the multiplication of decimals. 1) Carry out the multiplication as if there were no decimal places. 2) Count the number of decimal places in the sum. 3) Put this number of places in the answer. Look at the example: 2.5 x 3 = 7.5 because 25 x 3 = 75 There is one decimal place in the sum, so there must be one decimal place in the answer too. Look at the next example, which involves two decimal numbers. 2.4 x 0.3 = ? and 24 x 3 = 72 There are two decimal places in the sum, so the answer is 0.72 Exercise 4A: Now try these:

1) 33.4 x 2 = 2) 27.2 x 3 = 3) 51.7 x 4 = 4) 60.6 x 2 = 5) 54.4 x 3 = 6) 36.2 x 4 = 7) 50.3 x 7 = 8) 62.4 x 6 = 9) 39.2 x 7 = 10) 72.6 x 4 = 11) 38.4 x 7 = 12) 42.2 x 9 = Check Your Answers Look at the next example.

But this can’t be true, because we’ve only doubled a number that’s slightly bigger than 23, so the answer should be somewhere near 50. Since there are 2

places of decimals in the original sum, there must be 2 places of decimals in the answer. This means the answer is 53.82. This is reasonable, because 23.4 is nearly 24 and 2.3 is bigger than 2, and our answer is a bit bigger than 24 x 2 = 48 Exercise 4B: Now try these: 1) 3.6 x 1.4 = 2) 8.4 x 2.3 = 3) 12.2 x 3.9 = 4) 25.7 x 5.7 = 5) 45.3 x 4.2 = 6) 88.4 x 6.6 = 7) 3.34 x 2 = 8) 2.72 x 3 = 9) 5.17 x 4 = 10) 6.06 x 2 = 11) 5.04 x 3 = 12) 3.52 x 4 = 13) 5.13 x 7 = 14) 3.34 x 6 =

15) 3.28 x 7 = 16) 81.6 x 0.4 = 17) 38.4 x 0.7 = 18) 40.2 x 0.9 = 19) 72.6 x 0.4 20) 80.4 x 0.7 = 21) 33.2 x 0.9 = 22) 32.6 x 0.04 = 23) 98.4 x 0.07 = 24) 42.2 x 0.09 = Check Your Answers Look at the following example: 34.2 x 0.02 = ? If we estimate the value of this calculation we can say it is approximately equal to 34 x , but of 34 equals 0.34 and so our answer must be approximately equal to 0.68. Carrying out the calculation we have:

(Notice that zeros in front of a number do not affect the value of that number.) From our estimation we have the value of 34.2 x 0.02 = 0.684. This shows that the rule stated above holds good for all decimals, namely: The rule for multiplication of decimals. 1) Carry out the multiplication as if there were no decimal places. 2) Count the number of decimal places in the sum. 3) Put this number of places in the answer. Exercise 4C: Estimate the value of these calculations and then find the true results. 1) 24.1 x 0.04 = Approximation:

2) 34.9 x 0.05 = Approximation:

3) 35.6 x 0.02 = Approximation:

4) 14.7 x 0.06 = Approximation:

Sometimes a calculation will produce an answer like the one below.

Note: This calculation ends in a zero. But if we estimate the answer we find it to be larger than 24 x 4 = 96, and from the calculation above we can see that the answer is 113.490, which is the same as 113.49 (We can leave the zero off in our final answer, but we must keep it in place until we have added all the number places after the decimal point.) To make sure you have the right value, carry out an approximation to check your answer.

5) 31.72 x 3.5 = Approximation:

6) 44. 55 x 6.4 = Approximation:

Remember that zeros placed at the front of a whole number do not affect its value. For example, 00094 has the same value as 94. Likewise, zeros placed at the end of the decimal part of a number do not affect the value of the number. For example, 23.4400000 has the same value as 23.44. The zeros are purely used as place-keepers. 7) 60.02 x 1.5 = Approximation:

8) 14.05 x 11.4 = Approximation =

9) 40.07 x 2.5 = Approximation:

10) 10.05 x 12.4 = Approximation =

11) 57.02 x 1.5 = Approximation:

12) 19.05 x 11.6 = Approximation =

Check Your Answers We can change any decimal into a whole number by multiplying it by multiples of 10. For example: 3.01 x 100 = 301 Bonus Round: Change the following decimals into whole numbers. 1) 4.44 x 100 = 2) 12.72 x 100 =

3) 54.38 x 100 = 4) 65.41 x 100 = 5) 223.79 x 100 = 6) 104.30 x 100 = 7) 94.4 x 10 = 8) 10.7 x 10 = 9) 50.3 x 10 = 10) 55.42 x 100 = 11) 22.75 x 100 = 12) 79.38 x 100 = 13) 98.02 x 10 = 14) 90.05 x 10 = 15) 70.08 x 10 = Check Your Answers

Chapter 5: Division of Decimals Let us now look at the division of decimals. This is possibly the most difficult area to understand. However, there is a simple way to calculate them. Look at the division of numbers below.

Although the numbers in each calculation are different, the answers have all turned out the same. Can you see what we’ve done? Look at the next examples.

So here’s the rule: If you multiply the value of the number on both sides of the ÷ sign, the answer will always remain the same. Multiplying 14 and 7 by 3 will give you 42 and 21, and both 14 ÷ 7 and 42 ÷ 21 equal 2. The important thing is to multiply both sides of the ÷ sign by the same number. Try some for yourself. Exercise 5A: Fill in the missing numbers.

Check Your Answers Did you know that we can change any decimal into a whole number simply by multiplying it by a multiple of 10? Here is an example: 3.1 x 10 = 31 Exercise 5B: Change the following decimals to whole numbers. 1) 4.4 x 10 = 2) 12.7 x 10 = 3) 54.3 x 10 = 4) 65.4 x 10 = 5) 223.7 x 10 = 6) 104.3 x 10 = 7) 94.4 x 10 =

8) 10.7 x 10 = 9) 50.3 x 10 = 10) 55.42 x 100 = 11) 22.75 x 100 = 12) 79.38 x 100 = 13) 98.02 x 10 = 14) 90.05 x 10 = 15) 70.08 x 10 = Check Your Answers Since there is a close correlation between decimals and fractions, we could look at this process in a different way. We know that when any number is divided by one the answer always remains the same. So we need to make the right-hand-side equal to one.

If we now convert this statement into its decimal form, 0.25 Multiplying each side by 100 we get the expression:

÷

becomes: 0.5 ÷

(0.5 x 100) ÷ (0.25 x 100) = 50 ÷ 25 = 2 As you can see, we have arrived at the same answer. So here is a simple rule when dividing decimals: 1) Multiply both decimals on either side of the division sign by the same multiples of 10 (I.e. 10, 100, 1000, etc.) until the divisor is a whole number. 2) Divide in the same way as if you were carrying out a normal division sum. Example: 0.4 ÷ 0.2 = (0.4 x 10) ÷ (0.2 x 10) = 4 ÷ 2 = 2 Exercise 5C: Try these setting these out as above. 1) 0.5 ÷ 0.2 = 2) 0.6 ÷ 0.4 = 3) 0.7 ÷ 0.4 = 4) 0.6 ÷ 0.5 = 5) 0.8 ÷ 0.4 = 6) 0.9 ÷ 0.3 = 7) 1.5 ÷ 0.2 = 8) 2.4 ÷ 0.3 = 9) 1.4 ÷ 0.5 = 10) 6.4 ÷ 0.8 = 11) 10.4 ÷ 0.5 =

12) 32.8 ÷ 0.8 = 13) 20.5 ÷ 0.5 = 14) 60.8 ÷ 0.8 = 15) 24.4 ÷ 0.4 = 16) 30.1 ÷ 0.7 = Check Your Answers Now for examples using double-digit decimals. This time the divisor has 2 places of decimals. Therefore, we must multiply both sides by 100. Example: 0.4 ÷ 0.02 = (0.4 x 100) ÷ (0.02 x 100) = 40 ÷ 2 = 20 Exercise 5D: Now try these: 1) 24.44 ÷ 0.04 = 2) 30.17 ÷ 0.07 = 3) 24.33 ÷ 0.03 = 4) 30.17 ÷ 0.05 = 5) 124.2 ÷ 0.03 = 6) 200.15 ÷ 0.05 = 7) 24.01 ÷ 0.07 = 8) 36.09 ÷ 0.09 =

9) 20.46 ÷ 0.06 = 10) 56.07 ÷ 0.07 = 11) 32.16 ÷ 0.16 = 12) 54.27 ÷ 0.27 = 13) 200.25 ÷ 0.25 = 14) 99.33 ÷ 0.11 = Check Your Answers Notice that some of the answers are large. This is because you’re dividing a fairly large number by a very small one.

Exercise 5E: Fill in the missing decimals on the number lines below.

Check Your Answers We can extend the grid as many columns as we like in both directions. The value of each column decreases by 10 times as it moves to the right, and increases its value 10 times as it moves to the left.

In chapter 2 we changed a fraction into a decimal simply by dividing the numerator by the denominator (i.e. dividing the top number by the bottom one)

Exercise 5F: Find the decimal equivalents to the following common fractions. Use the

example above as a guide to setting out each calculation. 1)

=1÷8

2)

=3÷8

3)

=5÷8

4)

=7÷8

Recurring or Repeating Decimals

As mentioned before, some fractions are special. For example:

5)

=1÷6=

=1÷3

6)

=2÷3=

7)

=7÷9=

8)

=8÷9=

9)

= 5 ÷ 11 =

10)

= 1÷7=

What do you notice about the equivalent decimal of ?

11)

= 1 ÷ 9=

12)

= 2 ÷ 9=

What do you notice about the equivalent decimal of

and ?

Check Your Answers For this next exercise you’ll need a calculator. Use it to check your answers, and notice that the calculator will give you an answer with a large number of decimal points. This will enable you to see the recurring elements of each

fraction. Bonus Round: 1) Try changing

to a decimal.

2) Try changing

to a decimal.

3) Try changing

to a decimal.

4) Try changing

to a decimal.

5) Try changing

to a decimal.

6) Try changing

to a decimal.

7) Try changing

to a decimal.

8) Try changing

to a decimal.

9) Try changing

to a decimal.

10) Try changing

to a decimal.

11) Try changing

to a decimal.

Check Your Answers

Conclusion Part One By this point you should have a solid understanding of decimals and their nature. From decimal addition, subtraction, multiplication and division, you should also be able to understand decimals on the number line, how decimals relate to fractions, and how decimals work in number grids. As you progress in mathematics, you’ll encounter many more of these decimals. Regardless of how many decimal points they have, or whether they apply to currency or real-world problems, the fundamental properties underpinning them remain the same. Decimals and fractions are both closely related, being almost interchangeable in some cases. But these are not the only two kinds of number that are closely linked. There are, in fact, three. The third kind of number are percentages. In book two, you’ll discover how percentages relate to fractions, how they can be used to represent portions of whole numbers, and the important ways that they can be used in real-world problems. If you struggled with any of the areas in the decimals book, it is encouraged you pursue these further. Practice is paramount to success in mathematics, and focusing on your weaknesses is the key to proficiency. Hearing multiple different explanations of concepts is sometimes helpful, as is going through additional problems.

Decimals Answers Chapter 1 Back to Chapter 1 Exercise 1A 1) 469,351 - 74,030 - 604,403 - 80,059 - 330,482 2) 30,070 3) 175,200 4) 640,503 5) 88,017 6) 302,207 7) 801,519 8) 55,010 9) 1003 10) 288,500 Exercise 1B 1) 300+80+2+7/10 2) 400+70+5+3/10 3) 700+70+4+6/10 4) 600+0+5+8/10 5) 400+0+0+3/10 6) 600+70+8+8/10 7) 900+80+6+9/10 8) 300+0+2+6/10 9) 100+30+7/10 10) 20+5+8/10 11) 80+1+4/10 12) 50+5/10 Exercise 1C 1) 300+30+8+2/10+7/100 2) 600+60+7+5/10+6/100 3) 300+0+4+4/10+5/100

4) 800+0+0+8/10+7/100 5) 700+70+3+9/10+2/100 Exercise 1D 1) 500+70+6+27/100 2) 500+40+9+56/100 3) 300+0+4+45/100 4) 600+0+2+87/100 5) 300+20+9+92/100 6) 400+40+4+9/100 7) 400+0+8+7/100 8) 400+0+0+9/100 9) 600+0+4+5/100 10) 700+0+2+8/100 11) 600+20+9+1/100 Exercise 1E 1) 58,000+200+30+4+7/10 2) 73,000+400+9/10 3) 103,000+600+10+9+7/10 4) 10,000+100+30+2+4/10 5) 29,000+4/10 6) 203,000+30+4+1/10 7) 99,000+20+9+5/10 8) 23,000+300+8+2/10+7/100 9) 49,000+30+3/10+4/100 10) 60,000+200+90+8/10+2/100 11) 903,000+9+4/100 Exercise 1F 1) 40+5+2/10+6/100+4/1000 2) 60+2/10+3/100+5/1000 3) 90+8+4/10+5/100+6/1000 4) 40+8+6/10+7/100+9/1000 5) 900+9+8/10+9/100+4/1000

6) 100+40+7/100+3/1000

Chapter 2 Back to Chapter 2 Exercise 2A 1) 27/100 2) 14/100 3) 4/1000 4) 12/1000 5) 104/1000 6) 111/1000 Exercise 2B 1) 0.1 2) 0.3 3) 0.7 4) 0.2 5) 0.9 6) 0.4 7) 0.125 8) 0.375 9) 0.625 10) 0.091 Exercise 2C 1) 5.7 2) 9.9 3) 4.57 4) 9.59 5) 7.72 6) 9.65 7) 5.95 8) 10.49 9) A=5.2, B=5.5, C=6.1, D=6.4, E=6.5, F=6.8 10) A=6.8, B=7.2, C=7.9 11) A= 5.5, B=6.4, C=6.9 Exercise 2D 1) 0.11 2) 0.17 3) 0.34 4) 0.39 5) 0.71 6) 0.99 7) 0.01

8) 0.03 9) 0.06 10) 0.08 11) 0.09 12) 0.07 13) 4/10 14) 9/10 15) 7/10 16) 31/100 17) 8/10 18) 7/10 19) 25/10 20) 74/10 21) 83/10 22) 618/100 23) 983/100 24) 269/100 25) 393/100 26) 1097/100 Bonus Round: 1) 42/100 2) 96/100 3) 77/100 4) 7/100 5) 9/100 6) 4/100 7) 112/1000 8) 239/1000 9) 302/1000 10) 231/1000 11) 39/1000 12) 9/1000 13) 409/1000 14) 759/1000

Chapter 3 Back to Chapter 3 Exercise 3A 1) 765.8 2) 673.37 3) 699.9 4) 979.89 5) 518.786 6) 1240.381 7) 1433.967 8) 1790.584 Exercise 3B 1) 566.61 2) 697.83 3) 888.95 4) 899.94 5) 958.3 6) 730.5 Exercise 3C 1) 530.57 2) 297.77 3) 151.54 4) 353.62 5) 333.95 6) 291.43 7) 165.66 8) 112.87 9) 80.594 10) 85.517 11) 443.451 12) 221.034 13) 185.461 14) 72.083 Exercise 3D 1) 114.1 2) 434.1 3) 375.3 4) 214.12 5) 757.44 6) 152.02 7) 320.54 8) 433.41 9) 311.51 10) 322.21

11) 494.12 Exercise 3E 1) 164.37 2) 319.26 3) 389.29 4) 223.8 5) 422.9 6) 20.8 7) 400.5 8) 400.6 9) 440.7 Exercise 3F 1) 603.77 2) 321.84 3) 490.99 4) 422.69 5) 261.52 6) 570.64

Chapter 4 Back to Chapter 4 Exercise 4A 1) 66.8 2) 81.6 3) 206.8 4) 121.2 5) 163.2 6) 144.8 7) 352.1 8) 374.4 9) 274.4 10) 290.4 11) 268.8 12) 379.8 Exercise 4B 1) 5.04 2) 19.32 3) 47.58 4) 146.49 5) 190.26 6) 583.44 7) 6.68 8) 8.16 9) 20.68 10) 12.12 11) 15.12 12) 14.08 13) 35.91 14) 20.04 15) 22.96 16) 32.64 17) 26.88 18) 36.18 19) 29.04 20) 56.28 21) 29.88 22) 1.304 23) 6.888 24) 3.798 Exercise 4C 1) 0.964 Approx. 24 x 4/100 = 96/100 2) 1.745

Approx. 35 x 5/100 = 175/100 3) 0.712 Approx. 36 x 2/100 = 72/100 4) 0.882 Approx. 15 x 6/100 =90/100 5) 111.02 Approx. 32 x 4 = 120 6) 285.12 Approx. 45 x 6 = 270 7) 90.03 Approx. 60 x 2 = 120 8) 160.17 Approx. 14 x 11 = 154 9) 100.175 Approx. 40 x 3 =120 10) 124.62 Approx. 10 x 12 =120 11) 85.53 Approx. 57 x 2 = 114 12) 220.98 Approx. 20 x 11 = 220 Bonus Round 1) 444 2) 1272 3) 5438 4) 6541 5) 22379 6) 10430 7) 944 8) 107 9) 503 10) 5542 11) 2275 12) 7938 13) 980.2 14) 900.5 15) 700.8

Chapter 5 Back to Chapter 5 Exercise 5A 1) (9x4)÷(3x4)=36÷12 2) (15x4)÷(5x4)=60÷20 3) (4x10)÷(2x10)=40÷20 4) (12x10)÷(4x10) =120÷40 5) (9x10)÷(3x10)=90÷30 6) (12x10)÷(3x10)=120÷30 Exercise 5B 1) 44 2) 127 3) 543 4) 654 5) 2237 6) 1043 7) 944 8) 107 9) 503 10) 5542 11) 2275 12) 7938 13) 980.2 14) 900.5 15) 700.8 Exercise 5C 1) 2.5 2) 1.5 3) 1.75 4) 1.2 5) 2 6) 3 7) 7.5 8) 8 9) 2.8 10) 8 11) 20.8 12) 41 13) 41 14) 76 15) 61 16) 43 Exercise 5D 1) 611 2) 431

3) 811 4) 603.4 5) 4140 6) 4003 7) 343 8) 401 9) 341 10) 801 11) 201 12) 201 13) 801 14) 903 Exercise 5E 1) A 0.28, B 0.37 2) C 1.34, D 1.47 3) A 2.74, B 2.82, C 2.88 4) D 10.47, E10.58 5) G 9.93, H 10.04 Exercise 5F 1) 0.125 2) 0.375 3) 0.625 4) 0.875 5) 0.1666 6) 0.666 7) 0.777 8) 0.888 9) 0.4545 10) 0.142857142857 11) 0.1111 12) 0.222 Bonus Round 1) 0.142857 2) 0.1111 3) 0.083333 4) 0.090909 5) 0.9 6) 0.416666 7) 0.05 8) 0.583333 9) 0.545454 10) 0.833333 11) 0.888888

Part Two: Percentages

Percentages - Introduction The word "Percent" comes from the Latin Per Centum. The Latin word Centum means 100 – for example, a Century is 100 years. The term ‘per cent’ means ‘out of a hundred’. In mathematics, percentages are used to describe parts of a whole – the whole is made up of a hundred equal parts. The percentage symbol % is commonly used to show that the number is a percentage. So 1% is read as ‘one percent’, 5% as ‘five percent’, and so on. Percentages are ways of dividing the whole into 100 equal parts. The whole can be anything – an amount of money, a length of time, the weight of an object – the whole is simply the entire amount of something. Since percentages are ways of dividing the whole into 100 equal parts, they’re closely related to fractions. In fact, any percentage can be represented in fractional form. Likewise, they can also be converted into decimals. Decimals, Fractions, and Percentages are just different ways of expressing the same value. The key facts to remember are that 100% is equal to one ‘whole’ and that all percentages less than 100% are equivalent to a fraction of ‘one whole’.

Look at this grid. It has 100 squares.

So each square represents ‘one-hundredth of the whole’. We can say that each square represents 1% of the whole, or just 1%. It follows that 2 squares represents 2% of the whole, 3 squares 3% of the whole, and so on up to all 100 squares – in other words, 100% of the whole. In fractional terms this equates to:

1% =

, 2% =

, 3% =

, and so on up to

.

For this next exercise, assume that we’re working with a 100 square by 100 square grid, like the one above.

Exercise 0A: Complete the questions below.

6) What do you notice about the grid with 5 rows? _________________ 7) What percentage do 6 rows represent? ____________

8) What percentage do 7 rows represent? ____________

9) What percentage do 8 rows represent? ____________

10) What percentage do 9 rows represent? ____________

Check Your Answers Now for some new questions. Exercise 0B: Complete the following: 1) 10 squares represent: 2) 20 squares represent: 3) 30 squares represent: 4) 40 squares represent: 5) 50 squares represent: 6) 60 squares represent: 7) 70 squares represent: 8) 80 squares represent: 9) 90 squares represent: 10) 100 squares represent:

11) In the grid, there are ______ squares

How many squares are coloured? _________ What fraction of the whole is coloured? ________ 12) In the grid, there are ______ squares

How many squares are coloured? _________ What fraction of the whole is coloured? ________ 13) In the grid, there are ______ squares

How many squares are coloured? _________ What fraction of the whole is coloured? ________ 14) In the grid there are _______squares

How many squares are coloured? _________ What fraction of the whole is coloured? ________ 15) In the grid below, there are ______ squares

How many squares are coloured? _________ What fraction of the whole is coloured? ________ Check Your Answers The grid below has 10 squares coloured red, and we would write this as the fraction . But if we look carefully, we can see that we could make 9 more columns the same as the first one.

This means that

=

, and so it follows that

= 10%

Now look at the next example. The square has 2 columns coloured.

As a fraction of the whole we write

But

= , and so

= 20%

=

= 20%

Exercise 0C: Now try these:

Check Your Answers

All percentages are part of the number system, and therefore each percentage occupies a unique position on the number line. This number line below shows the correlation between tenths and their percentage equivalent.

From this line, you can see that some percentages can be expressed as simplified fractions.

From the line, we can see that 50% = . It follows then that 25%, and therefore

of 50% must be

= 25%. The line now looks like this.

We can find a percentage of a group. Consider £1 = 100 pence (or $1 = 100 cents for Americans). If we place these values on a number line we get:

Chapter 6: Fractions and Percentages There is a close correlation between fractions and percentages, and so, it’s time we look for more fractional equivalents. Look at the grid. How many squares are coloured? Write your answer as a fraction in its lowest terms.

You should have found from the example above that,

It follows, then, that

=

=

= 25%.

= 75% and so on.

So now we can see that any percentage can be written as a fraction with the denominator of 100.

Exercise 6A: Write beside the grids the percentage coloured.

Check Your Answers

Just to confirm, all percentages are written in the form number, so that X% =

where X is any

and vice versa.

To enable us to find the percentage of a number, price, or other defined amount, we need to be able to convert any percentage in the form into a fraction in its lowest form. These are the most common forms. Learn them by heart so that you recognize them instantly.

Now we must look at the reason why. For example why is 40% equal to the fraction ? If we take any fraction and multiply it by 1, the answer remains the same. But 1 can be written as a fraction like this: , , , and so on. We also found that any fraction that has 100 as its denominator is easily turned into a percentage. Look at the examples below.

We have simply multiplied each fraction by one in the form

= 1.

Now let us look at other fractions that do not have denominators of 10.

We have found that 50%.

= 50%. We arrive at this by stating that

x

=

We simply found a fraction with a value equal to 1 that would change the denominator of our fraction to 100. Exercise 6B: Complete the following using the method above.

1) Convert

into a percentage.

2) Convert

into a percentage.

3) Convert

into a percentage.

4) Convert

into a percentage.

=

5) Convert

into a percentage.

6) Convert

into a percentage.

7) Convert

into a percentage.

8) Convert

into a percentage.

9) Convert

into a percentage.

10) Convert

into a percentage.

Check Your Answers We must now look at the reverse of this process. Look at the next example.

40% =

So 40% =

but

=

and we know that

(from above), and 40% also equals

= 1.

x

, so 40% is equal to

. We have simply divided the top and bottom by 20. This is known as cancelling down. There will be more on that in a minute.

Exercise 6C: Now try these:

1) 60% =

=

2) 80% =

=

3) 70% =

=

4) 90% =

=

5) 25% =

=

6) 75% =

=

7) 30% =

=

8) 20% =

=

9) 10% =

=

10) 5% =

=

11) 15% =

=

12) 35% =

=

Check Your Answers Cancelling Down So far, we have divided the top and bottom of fractions by 25, 20, 10 and 5, but 2 and 4 are also factors of 100, and therefore we can use them to reduce other percentages to fractions in their lowest terms. Earlier, we learnt that every percentage can be expressed as a fraction over 100, and that quite often we can reduce these fractions to more recognizable ones. We can use a technique called cancelling down, which works like this: Whenever there is a common factor in the number on the top and on the bottom of a fraction, they cancel each other out. Look at this example:

. This can be written as

is the same as

x , but

= 1 and so

= , which means that 50 is the common factor. We cancelled down using the fact that 50 is a common factor of both 50 and 100. We write

=

We will examine this method further. Here is another example:

20% =

=

x

= 1 x , and so 20% =

Cancelling down looks like this: both 20 and 100.

You can cancel this fraction down: 25 and 100.

because 20 is a common factor of

=

because 25 is a factor of both

You can cancel down any fraction as long as that number is a factor of both the top and bottom of the fraction. Look at the next set of questions. To reduce any percentages to their lowest terms, we must find a number that is a common factor of both the numerator and the denominator. Exercise 6D: Now try these. Remember to always use the largest common factor to cancel down the top and bottom of your fraction.

1) 12% =

=

2) 16% =

=

3) 24% =

=

4) 36% =

5) 2% =

=

=

6) 18% =

=

7) 28% =

=

8) 34% =

=

9) 5% =

=

10) 45% =

=

11) 55% =

=

12) 65% =

=

13) 85% =

=

14) 66% =

=

15) 74% =

=

16) 84% =

=

Check Your Answers

Chapter 7: Percentages of Whole Numbers Why then do we reduce percentages to fractions in their lowest form? The answer is that it’s easier to find a simple fraction of a number, monetary value or other objects than it is to find a percentage in the form of something different to 100. Look at the example: What is 25% of 16?

This might look difficult, but we know from previous work that 25% = , and then

of 16 = 4

Exercise 7A: Calculate the following: 1) 50% of 36 = 2) 20% of 35 = 3) 25% of 20 = 4) 10% of 50 = Check Your Answers We can simplify other percentages like this:

30% of 20 = Now

=

.

of 20 = 2, Therefore

of 20 = 3 x 2 = 6

Exercise 7B: Calculate the following: 1) 30% of 30 = 10% of 30 = Therefore 30% of 30 = 2) 30% of 60 = 10% of 60 = Therefore 30% of 60 = 3) 30% of 80 = 10% of 80 = Therefore 30% of 80 = 4) 30% of 90 = 10% of 90 = Therefore 30% of 90 = 5) 40% of 20 = 10% of 20 = Therefore 40% of 92 = 6) 40% of 30 = 10% of 30 = Therefore 40% of 30 = 7) 40% of 40 = 10% of 40 = Therefore 40% of 40 = 8) 40% of 60 = 10% of 60 = Therefore 40% of 60 = 9) 60% of 40 = 10% of 40 =

Therefore 60% of 40 = 10) 70% of 30 = 10% of 30 = Therefore 70% of 30 = 11) 20% of 110 = 10% of 110 = Therefore 20% of 110 = 12) 20% of 130 = 10% of 130 = Therefore 20% of 130 = 13) 30% of 120 = 10% of 120 = Therefore 30% of 120 = 14) 30% of 160 = 10% of 160 = Therefore 30% of 160 = 15) 40% of 120 = 10% of 120 = Therefore 40% of 120 = 16) 40% of 130 = 10% of 130 = Therefore 40% of 130 = 17) 60% of 110 = 10% of 110 = Therefore 60% of 110 = 18) 70% of 120 = 10% of 120 = Therefore 70% of 120 =

Check Your Answers

We found that 5% =

=

It follows that of a number equals the total number divided by 20. To simplify the process we can simply divide by 10 and then divide by 2. Look at the example: 5% of 200 = 10% of 200 = 200 ÷10 = 20. Therefore 5% of 200 = 20 ÷ 2 = 10. Exercise 7C: Now try these: 1) 5% of 100 = 2) 5% of 80 = 3) 5% of 60 = 4) 5% of 40 = Because 5% is half of 10% we can find 5% of more difficult numbers. For example: 5% of 24 = 10% of 24 = 24 ÷ 10 = 2.4 Therefore 5% of 24 = 2.4 ÷ 2 = 1.2 Now try these: (Don’t forget to set each one out as above.)

5) 5% of 48 = 6) 5% of 52 = 7) 5% of 64 = 8) 5% of 84 = 9) 5% of 96 = 10) 5% of 104 = 11) 5% of 122 = 12) 5% of 144 = Check Your Answers Now that we can find 5% of a number we can move to finding 15% of a number. Study the example below. We know that 15% = 10% + 5%. This will enable us to solve the problem. Find 15% of 40. 10% of 40 = 4 5% of 40 = 4 ÷ 2 = 2 15% 0f 40 = 4 + 2 = 6 Exercise 7D: Complete the following. 1) 15% of 60 = 2) 15% of 80 = 3) 15% of 20 =

4) 15% of 24 = 5) 15% of 48 = 6) 15% of 64 = 7) 15% of 90 = 8) 15% of 104 = 9) 15% of 66 = 10) 15% of 84 = 11) 15% of 72 = 12) 15% of 96 = 13) 15% of 108 = 14) 15% of 124 = Check Your Answers The previous method can be used to find other percentages. For example, suppose we want to find 45% of a number. We proceed like this: 45% = 10% + 10% + 10% + 10% + 5%. Study the example: 45% of 40 = 10% of 40 = 4 40% of 40 = 4 x 4 =16 5% of 40 = 2 Therefore 45% of 40 = 16 + 2 = 18 Exercise 7E: Now try these:

1) 55% of 60 = 2) 55% of 40 = 3) 55% of 70 = 4) 55% of 80 = 5) 55% of 30 = 6) 55% of 64 = 7) 55% of 120 = 8) 55% of 160 = 9) 35% of 70 = 10) 35% of 80 = 11) 35% of 60 = 12) 35% of 20 = 13) 35% of 90 = 14) 35% of 50 = 15) 35% of 10 = 16) 35% of 120 = Try finding a percentage of the numbers in these examples. 17) What is 50% of 20? 18) What is 50% of 30?

19) What is 20% of 20? 20) What is 20% of 30? 21) What is 10% of 20? 22) What is 10% of 30? 23) What is 25% of 20? 24) What is 25% of 40? 25) What is 50% of 44? 26) What is 50% of 60? 27) What is 20% of 60? 28) What is 20% of 50? 29) What is 10% of 40? 30) What is 10% of 60? Check Your Answers The numbers used in all of these examples could have been any number of items, say 40 dogs, or 20 pupils – the calculations are the same. We now have another way of finding a percentage of any number. Look at the example:

We have used the expression ‘of’ in this case to mean multiply. This method is well suited to using a calculator. Exercise 7F: Using the method above, find the following values. You can use a long multiplication method or a calculator. 1) 8% of 12 = 2) 13% of 13 = 3) 21% of 16 = 4) 27% of 18 = 5) 32% of 20 = 6) 38% of 24 = 7) 43% of 26 = 8) 46% of 32 = 9) 55% of 40 = 10) 67% of 42 = 11) 71% of 30 = 12) 77% of 48 = 13) 90% of 40 = 14) 97% of 54 = Check Your Answers

So far we have found the more common percentages and their equivalent fractions. But since percentages are part of the number system, there is a percentage equivalent to every fraction and, therefore, since there is an infinite number of fractions, it follows that there is an infinite number of percentages. We can calculate some more common ones using the information we already have. For example:

If

Since

= 10%, what is the percentage equivalent to

of

=

then

must be equal to

?

of 10%, which is 5%

Bonus Round: Now try these:

1) If

= 30% what is the fractional equivalent to 15%?

2) If

= 70% what is the fractional equivalent to 35%?

3) If

= 90% what is the fractional equivalent to 45%?

4) If

= 5% what percentage does

equal?

5) If

= 5% what percentage does

equal?

Every fraction has a percentage value. To calculate some additional percentages, we can use some of the simple values to calculate the percentage of more complex values. Study the following example:

We know that us to find

= 25%. We know also that

as a percentage.

= 25% therefore

as a percentage equals 25 ÷ 2 = 12 %

6) What is the value of

as a percentage?

7) What is the value of

as a percentage?

8) What is the value of

as a percentage?

9) If

10) If

is half of a quarter. This allows

= 25% what is the percentage equivalent to ?

= 75% what is the percentage equivalent to ?

What about thirds? First, we must find one-third of one hundred. i.e. 100 ÷ 3 = 33r1 (33 with a remainder of 1) which converts to 100 ÷ 3 = 33

So we have

= 33 % or alternatively written as 33.3%

11) What is

as a percentage?

12) What is

as a percentage?

13) What is

as a percentage?

Check Your Answers

Chapter 8: Percentages in Problems At this point, you should be familiar with the idea that percentages and fractions are closely linked. You will see adverts that offer a discount of 30%, or sale items that are reduced by 20%. What these statements mean is that the price of some goods has been reduced by a percentage of their original price. Let us take a concrete example. Suppose we buy a computer that has been reduced by 20% of its original price of $200. We must first find the value of 20% of $200.

We know that 20% = which equals $40.

= , so 20% of $200 is the same as

of $200,

Now, to find the new price, we simply say $200 - $40 = $160, so the reduced price is now $160 and you will have saved $40 on the deal. Exercise 8A: Find the new price of the following a 20% reduction in price. Set each one out as above. 1) $2.50 2) $3.75 3) $10.2 4) $12.70 Find the new value of the following sums of money after a 10% reduction in price. 5) $4.20

6) $6.20 7) $8.90 8) $11.20 9) $16.30 10) $21.10 Find the new value of the following sums of money after a 25% reduction in price. 11) $13.20 12) $39.20 13) $64.80 14) $380.00 15) $108.00 16) $276.40 17) $184.12 18) $156.28 19) $240.64 Find the new value of the following sums of money following a 15% reduction in price. Don’t forget to first find 10% and then 5%. 20) $20.00 21) $15.00

22) $24.00 23) $38.00 24) $62.00 25) $104.00 26) $82.00 27) $51.60 28) $58.20 29) $70.20 Check Your Answers A value can also be increased by a given percentage. This implies that you increase the original price of an object by a given percentage. To find the following 25% increase we proceed as in the example below.

25% of $2.40 =

of $2.40 = $2.40 ÷ 4 = 60 cents.

The new price will be $2.40 + 60 cents = $3.00 Exercise 8B: Find the new price following a 20% increase in price. Set each one out as above. 1) $2.50 2) $3.75

3) $10.25 4) $12.70 5) $15.25 Find the new value of the following sums of money after a 10% increase in price. 6) $4.20 7) $6.20 8) $8.90 9) $11.20 Find the new value of the following sums of money after a 25% increase in price. 10) $13.20 11) $39.20 12) $64.80 13) $380.00 14) $108.00 15) $220.00 Find the new value of the following sums of money following a 15% increase in the price. Don’t forget to first find 10% and then 5%. 16) $20.00

17) $15.00 18) $24.00 19) $38.00 20) $51.60 21) $70.20 Check Your Answers Percentages are involved in many real life scenarios. These next questions should illustrate that. Exercise 8C: Complete the following questions. Make sure you show your workings-out. 1) A bicycle costing $240 is reduced in a sale by 25%. What is the sale price of the bicycle? 2) A school with 360 pupils sees that number increase by 10% in the next year. How many pupils will be at the school next year? 3) David gets $3.50 per week. On his next birthday that will be increased by 20%. How much will he receive each week? 4) 7 out of every 10 people questioned liked a certain brand of cereal. What is this as a percentage? 5) If, in a test, you gained 25 marks out of 100, what percentage does your score represent? 6) Of the 200 visitors to the cinema, 40% were children. How many were adults and how many were children?

7) A man cycles 75 kilometres and finds that he has completed 30% of his journey. How much further does he have to go to complete his journey? 8) Calculate 30% of $250. 9) One-fifth of a class of 30 failed an English exam. How many passed the exam?

10)

of a group of 64 students were male. How many were female?

11) A man weighing 96 kilograms lost 12 % of his weight. How much does he weigh now? 12) I have 25 marbles. I lose 20% of them. How many marbles do I have left? 13) 1 large pizza costs $10.00. If I buy 3 of them I get 25% off the total bill. How much should I pay? 14) Michael is 190cm tall, and his sister Sarah is 10% shorter. How tall is Sarah? 15) Peter sold 300 tickets to a disco. Only 89% of people turned up. How many were at the disco? 16) A carpenter needs to cut a plank of wood that is 3.75m long into 5 equal pieces. What percentage of the plank is each piece? 17) A television costs $456 before a sale. In the sale, prices are reduced by 25%. How much does the TV cost in the sale? 18) Last year, Mr. Smith weighed 80kg. This year he weighs 17% more. How much does Mr. Smith weigh now? 19) The total attendance at last year's inter-school football final was 8400. This year's attendance fell by 6%. How many people went to this year’s

final? 20) Sammy saves $2.50 a week for 12 weeks. He then spends 9% on a new toy. How much does he have left? 21) Mike and his sister have 80 sweets between them. His sister eats 15% of the sweets and he eats 35%. (i) How many sweets has his sister eaten? (ii) How many sweets has Mike eaten? 22) 55% of what value equals $44.55? 23) A car is going to be reduced by 12% if I pay for it in cash. If the car costs $13,500 before the reduction, how much will it cost to buy after the reduction? 24) Sara is 170cm tall, and her sister is 22% shorter. How tall is her sister? 25) David and Ryan collected a number of stamps. David collected 80 stamps, and found that he had collected 32% of the total in the book. Ryan collected 12% of the total. How many stamps were there in the book, and how many did Ryan collect? 26) 11% of what sum of money equals $13.31? 27) Value Added Tax (VAT) is charged at 20% on most goods. How much of a toy costing $16:40 is Tax? 28) A Skateboard is reduced by 20% in a sale. The old price was $48:00. What is the new price?

29) Shop A sold of its stock of balloons. Shop B sold 65% of its stock of balloons. Which shop sold the most balloons?

30) At a school meeting, there were 80 people in attendance. If 70% of them were male, how many females were at the meeting? 31) A Football Club had an election to select a captain. 60% of the 95 members of the club voted in the election. How many members voted? Check Your Answers

Conclusion Part Two Congratulations! You have finished this book! By this point you should have a solid understanding of percentages, along with how they work. You should be able to apply percentages knowledge to whole numbers, number lines, and real-world problems, as well as change percentages to and from fractions. In addtion to this, you should also have a clear understanding of decimals too. In the future, you may encounter percentages which are larger than 100%. A value, like an amount of money, might grow by 250%. You will also encounter percentages in the business world, in the form of interest, taxes, and insurance. Because of this, a sound knowledge of percentages is vital. But regardless of how complex the problems you may encounter become, the fundamentals underpinning them remain the same. If you struggled with any of the areas in the percentages book, it is encouraged you pursue these further. Practice is paramount to success in mathematics, and focusing on your weaknesses is the key to proficiency. Hearing multiple different explanations of concepts is sometimes helpful, as is going through additional problems. By now you have reached the end of this book, and completed over 550 questions. We hope that you have learned some new concepts, becoming confident in your math abilities, and are ready to move on to new and more difficult areas.

Percentages Answers Introduction Back to Percentages Introduction Exercise 0A 1) 10% 2) 20% 3) 30% 4) 40% 5) 50% 6) That it equals one half 7) 60% 8) 70% 9) 80% 10) 90% Exercise 0B 1) 10% 2) 20% 3) 30% 4) 40% 5) 50% 6) 60% 7) 70% 8) 80% 9) 90% 10) 100% 11) 100, 50, 1/2 12) 100, 5, 1/20 13) 100, 10, 1/10 14) 100, 20, 1/5 15) 100, 15, 3/20 Exercise 0C 1) 3/10 = 30% 2) 4/10 = 40% 3) 1/2 = 50%

4) 6/10 = 60% 5) 7/10 = 70% 6) 8/10 =80% 7) 9/10 = 90% 8) 100/100 =100%

Chapter 6 Back to Chapter 6 Misc.) 1/4 = 25% Exercise 6A 1) 15% 2) 24% 3) 42% 4) 14% 5) 34% 6) 62% Exercise 6B 1) 1/5 x 20/20 = 20/100 = 20% 2) 3/10 x 10/10 = 30/100 = 30% 3) 7/10 x 10/10 = 70/100 = 70% 4) 1/4 x 25/25 = 25/100 = 25% 5) 3/4 x 25/25 = 75/100 = 75% 6) 2/5 x 20/20 = 40/100 = 40% 7) 3/5 x 20/20 = 60/100 = 60% 8) 4/5 x 20/10 = 80/100 = 80% 9) 9/10 x 10/10 = 90/100 = 90% 10) 10/10 x 10/10 = 100/100 = 100% Exercise 6C 1) 20/20 x 2/5 2) 20/20 x 4/5 3) 10/10 x 7/10 4) 10/10 x 9/10 5) 25/25 x 1/4 6) 25/25 x 3/4 7) 10/10 x 3/10 8) 20/20 x 1/5 9) 10/10 x 1/10

10) 5/5 x 1/20 11) 5/5 x 3/20 12) 5/5 x 7/20 Exercise 6D 1) 4/4 x 3/25 2) 4/4 x 4/25 3) 4/4 x 6/25 4) 4/4 x 9/25 5) 2/2 x 1/50 6) 2/2 x 9/50 7) 4/4 x 7/25 8) 2/2 x 17/50 9) 5/5 x 1/20 10) 5/5 x 9/20 11) 5/5 x 11/20 12) 5/5 x 13/20 13) 5/5 x 17/20 14) 2/2 x 33/50 15) 2/2 x 37/50 16) 4/4 x 21/25

Chapter 7 Back to Chapter 7 Exercise 7A 1) 18 2) 7 3) 5 4) 5 Exercise 7B 1) 9 2) 18 3) 24 4) 27 5) 8 6) 12 7) 16 8) 24 9) 24 10) 21 11) 22 12) 26 13) 36 14) 48 15) 48 16) 52 17) 66 18) 84 Exercise 7C 1) 5 2) 4 3) 3 4) 2 5) 2.4 6) 2.6 7) 3.2 8) 4.2 9) 4.8 10) 5.2 11) 6.1 12) 7.2 Exercise 7D 1) 9 2) 12

3) 3 4) 3.6 5) 7.2 6) 9.6 7) 13.5 8) 15.6 9) 9.9 10) 12.6 11) 10.8 12) 14.4 13) 16.2 14) 18.6 Exercise 7E 1) 33 2) 22 3) 38.5 4) 44 5) 16.5 6) 35.2 7) 66 8) 88 9) 24.5 10) 28 11) 21 12) 7 13) 31.5 14) 17.5 15) 3.5 16) 42 17) 10 18) 15 19) 4 20) 6 21) 2 22) 3 23) 5 24) 10 25) 22 26) 30 27) 12 28) 10 29) 4 30) 6 Exercise 7F 1) 0.96 2) 1.69 3) 3.36 4) 4.86

5) 6.4 6) 9.12 7) 11.18 8) 14.72 9) 22 10) 28.14 11) 21.3 12) 36.96 13) 36 14) 52.38 Bonus Round 1) 3/20 2) 7/20 3) 9/20 4) 35% 5) 45% 6) 371/2% 7) 621/2% 8) 871/2% 9) 121/2% 10) 371/2% 11) 662/3% 12) 162/3% 13) 831/3%

Chapter 8 Back to Chapter 8 Exercise 8A 1) $2.00 2) $3.00 3) $8.16 4) $10.16 5) $3.78 6) $5.58 7) $8.01 8) $10.08 9) $14.67 10) $18.99 11) $9.90 12) $29.40 13) $48.60 14) $285.00 15) $81.00 16) $207.30 17) $138.09 18) $117.21 19) $180.48 20) $17.00 21) $12.75 22) $20.40 23) $32.30 24) $52.70 25) $88.40 26) $69.70 27) $43.86 28) $49.47 29) $59.67 Exercise 8B 1) $3.00 2) $4.50 3) $12.30 4) $15.24 5) $18.30 6) $4.62 7) $6.82 8) $9.79 9) $12.32 10) $16.50 11) $49.00

12) $81.00 13) $475.00 14) $135.00 15) $275.00 16) $23.00 17) $17.25 18) $27.60 19) $43.70 20) $59.34 21) $80.73 Exercise 8C 1) $240 - $60 = $180 2) 360 + 36 = 396 pupils 3) $3.50 -+ 70 cents = $4.20 4) 70% 5) 25% 6) 200 – 80 = 120 7) 100% = 250kms 8) $75 9) 30 – 6 = 24 pupils 10) 24 11) 96 – 12 = 84kgs 12) 20 13) $22.50 14) 171cms 15) 267 16) 20% 17) $342.00 18) 93.6kgs 19) 7896 20) $27.30 21) 12 & 28 sweets 22) $81.00 23) $11,880 24) 132.6cms 25) 250 stamps, Ryan 30 26) $121.00 27) $3.28 28) $38.40 29) B 30) 24 31) 57