00 found-prelims ii-vi - pearson education · 38 pythagoras’ theorem 573 38.1 pythagoras’...

ii

1 Collecting and recording data 1

1.1 Collecting data by observation and experiment 1

1.2 Questionnaires 51.3 Sampling 81.4 Databases 9Chapter review questions 15

2 Processing, representing and interpreting data 18

2.1 Pictograms 182.2 Bar charts 212.3 Pie charts 242.4 Using pie charts 272.5 Time series 312.6 Grouping data 332.7 Frequency polygons 35Chapter review questions 40

3 Averages and range 463.1 Mean, mode, median and range 463.2 Using frequency tables to find averages 50

3.3 Stem and leaf diagrams 533.4 Data logging and comparing

distributions 553.5 Estimating the mean of

grouped data 59Chapter review questions 63

4 Probability 664.1 The probability scale 664.2 Writing probability as numbers 684.3 Two-way tables 714.4 Sample space diagrams 744.5 Mutually exclusive outcomes

and the probability that the outcome of an event will not happen 76

4.6 Estimating probability fromrelative frequency 78

Chapter review questions 81

5 Scatter graphs 875.1 Scatter graphs and relationships 875.2 Lines of best fit and correlation 915.3 Using lines of best fit 92Chapter review questions 96

Contents

MODULE 2

6 Introducing number 1006.1 Numbers and place value 1006.2 Number lines 1026.3 Rounding numbers 1046.4 Mental methods 1066.5 Written calculations 1086.6 Solving problems with and

without a calculator 1136.7 Factors, multiples, squares and cubes 1176.8 Order of operations 1196.9 Writing a number as a product

of its prime factors 1226.10 Highest common factors and

lowest common multiples 124Chapter review questions 126

7 Angles 1 1297.1 Fractions of a turn and degrees 1297.2 What is an angle? 1297.3 Special types of angles 1307.4 Naming sides and angles 1307.5 Perpendicular lines and parallel

lines 1317.6 Estimating angles 1337.7 Measuring angles and lines 1347.8 Drawing angles and shapes 1377.9 Angle facts 138Chapter review questions 142Names of parts of a circle 145

�M2, 4

�M2, 4

�M4

�M4

MODULE 3

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8 Fractions and decimals 1468.1 What is a fraction? 1468.2 Equivalent fractions 1488.3 Simplifying fractions 1508.4 Ordering fractions 1518.5 Improper fractions and mixed numbers 1538.6 Reading and writing decimals 1558.7 Understanding place value 1578.8 Ordering decimals 1588.9 Converting decimals to

fractions 1608.10 Converting fractions to

decimals 162Chapter review questions 165

9 Directed numbers 1679.1 What is a directed number? 1679.2 Addition and subtraction of directed

numbers 1709.3 Multiplication and division of directed

numbers 1729.4 Using a calculator 173Chapter review questions 175

10 Decimals 17810.1 Rounding decimals 17810.2 Estimating 17910.3 Adding and subtracting decimals 18110.4 Multiplying decimals 18310.5 Dividing decimals 18710.6 Further estimates 19010.7 Rounding to decimal places 19110.8 Rounding to significant figures 19310.9 Problems with decimals 195Chapter review questions 197

11 Algebra 1 20011.1 Using letters to represent numbers 20011.2 Expressions and terms 20211.3 Collecting like terms 20511.4 Multiplying with numbers and

letters 20611.5 Multiplying out brackets 20811.6 Factorising 209Chapter review questions 211

12 Perimeter and area of 2-D shapes 212

12.1 Perimeter 21212.2 Area 21512.3 Areas of rectangles, squares, triangles,

and parallelograms 21912.4 Problems involving areas 223Chapter review questions 226

13 Sequences 23013.1 Sequences 23013.2 Input and output machines 23513.3 Finding an expression for the nth

term of an arithmetic sequence 240Chapter review questions 244

14 Angles 2 24714.1 Triangles 24714.2 Equilateral triangles and isoceles

triangles 24814.3 Corresponding angles and alternate

angles 25014.4 Proofs 25314.5 Bearings 254Chapter review questions 258

15 Graphs 1 26015.1 Coordinates in the first quadrant 26015.2 Coordinates in all four quadrants 26115.3 Finding the coordinates of the

midpoint of a line 262Chapter review questions 264

16 Graphs 2 26616.1 Equations of vertical and horizontal

lines 26616.2 Straight line graphs 26816.3 Graphs of x � y � k 272Chapter review questions 273

�M4

�M4

�M4

�M4

�M4

�M4

�M4

�M4

�M4

�M4

�M2

00 Found-Prelims ii-vi.qxd 8/6/06 9:41 am Page iii

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CONTENTS

17 Measure 27517.1 Reading scales 27517.2 Time 27917.3 Units 28717.4 Converting between metric units 28917.5 Converting between metric and

imperial units 29017.6 Compound measures – speed and

density 292Chapter review questions 297

18 Percentages 1 30318.1 Introduction 30318.2 Percentages of quantities 308Chapter review questions 310

19 Powers and roots 31219.1 Powers and roots 31219.2 Order of operations 31419.3 Index laws for multiplication and

division 31619.4 Using a calculator 317Chapter review questions 320

20 Three-dimensional shapes 322

20.1 Types of three-dimensional shapes 32220.2 Faces, vertices and edges 32320.3 Nets 32420.4 Isometric paper 32620.5 Volume 32720.6 Surface area 33320.7 Dimensions 33520.8 Coordinates in three dimensions 337Chapter review questions 339

21 Algebra 2 34221.1 Multiplying out brackets 34221.2 Further factorising 344Chapter review questions 345

22 Estimating and accuracy 34622.1 Rounding 34622.2 Solving problems using approximations 34922.3 Interpreting calculator displays 35122.4 Problem solving using a calculator 35322.5 Accuracy of measurements 355Chapter review questions 357

�M4

�M4

�M4

�M4

�M4

�M4

23 Two-dimensional shapes 1 359

23.1 Symmetry 35923.2 Triangles 36523.3 Symmetry of quadrilaterals 36523.4 Symmetry of regular polygons 36723.5 Drawing shapes 36823.6 Congruent shapes 37323.7 Circles 37523.8 Tessellations 377Chapter review questions 379

24 Angles 3 38624.1 Quadrilaterals 38624.2 Polygons 38924.3 Exterior angles 393Chapter review questions 395

25 Practical graphs 39725.1 Real-life graphs 39725.2 Conversion graphs 40225.3 Distance–time graphs 406Chapter review questions 409

26 Further fractions 41226.1 Fractions of an amount 41226.2 Expressing one amount as a fraction

of another 41426.3 Addition and subtraction of

fractions 41626.4 Addition and subtraction of mixed

numbers 41826.5 Multiplication of a fraction by an

integer 42026.6 Division of a fraction by an integer 42126.7 Multiplication of a fraction by a unit

fraction 42226.8 Multiplication of fractions 42326.9 Calculations with fractions 424Chapter review questions 426

27 Graphs 3 42827.1 Graphs of y � mx � c 42827.2 Graphs of quadratic functions 43127.3 Using graphs of quadratic functions

to solve equations 434Chapter review questions 437

MODULE 4

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CONTENTS

28 Percentages 2 43928.1 Using percentages 43928.2 One quantity as a percentage of

another quantity 442Chapter review questions 444

29 Equations and inequalities 446

29.1 Equations 44629.2 The balance method for solving

equations 44729.3 Setting up equations 45229.4 Solving equations that have

brackets 45429.5 Solving equations that have letters

on both sides 45629.6 Inequality signs 45829.7 Inequalities on a number line 45929.8 Solving inequalities 46029.9 Integer solutions to inequalities 461Chapter review questions 463

30 Formulae 46630.1 Using a word formula 46630.2 Writing a word formula 46830.3 Using an algebraic formula 47030.4 Writing an algebraic formula 47230.5 Using a formula inversely 47330.6 Changing the subject of a formula 47530.7 Expressions, identities, equations and

formulae 477Chapter review questions 478

31 The circle, the cylinder 48231.1 Circumference of a circle 48231.2 Area of a circle 48531.3 Circumferences and areas in terms of � 48731.4 Volume and surface area of a prism 489Chapter review questions 491

32 Converting units of area and volume 495

32.1 Converting units of areas 49532.2 Converting units of volume 496Chapter review questions 498

33 Ratio and proportion 49933.1 Introduction to ratio 49933.2 Problems on ratios 50433.3 Sharing a quantity in a given ratio 50633.4 Proportion 508Chapter review questions 511

34 Planes of symmetry,plans and elevations 515

34.1 Planes of symmetry 51534.2 Plans and elevations 516Chapter review questions 518

35 Transformations 51935.1 Introduction 51935.2 Translations 51935.3 Rotations 52535.4 Reflection 52835.5 Enlargement 53535.6 Centre of enlargement 540Chapter review questions 545

36 Further algebra 55036.1 Index notation 55036.2 Substituting negative numbers into

expressions involving powers 55236.3 Solving equations of the type ax2 � b 55336.4 Trial and improvement 554Chapter review questions 557

37 Constructions and loci 55937.1 Constructions 55937.2 Loci 56137.3 Regions 566Chapter review questions 570

38 Pythagoras’ theorem 57338.1 Pythagoras’ theorem 57338.2 Finding the length of a hypotenuse 57438.3 Finding the length of one of the

shorter side of a right-angled triangle 57638.4 Applying Pythagoras’ theorem 578Chapter review questions 582

Index 585

Licence Agreement 590

00 Found-Prelims ii-vi.qxd 8/6/06 9:41 am Page v

146

8C H A P T E R

Fractions and decimals

8.1 What is a fraction?

�27� of this circle is shaded.

�27� is a fraction.

Write down the fraction of the shape that is shaded.

Solution 13 parts are shaded.So the top number (numerator) of the fraction is 3

The circle is divided into 5 equal parts.So the bottom number (denominator) of the fraction is 5

�35� of the shape is shaded.

There are 30 students in a class.17 of the students walk to school.Write down the fraction of the students that

a walk to school b do not walk to school.

Solution 2a There are 17 students out of 30 that walk to school.

The fraction of students that walk to school is �1370�

b 30 � 17 � 1313 students out of 30 do not walk to school.

The fraction of students that do not walk to school is �1330�

Example 2

Example 1

27The bottom number shows that the

circle is divided into 7 equal parts.

The top number shows that 2 parts of the circle are shaded.

The bottom number of the fractionis called the denominator.

The top number of the fractionis called the numerator.

08 Chapter 08 146-166.qxd 1/6/06 11:40 am Page 146

147

8.1 What is a fraction? CHAPTER 8

Exercise 8A

In questions 1 to 8, write down the fraction of the shape that is shaded.

1 2 3 4

5 6 7 8

9 Write down the fraction of the shape that is

a shaded b unshaded.

10 Write down the fraction of the shape that is

a shaded b unshaded.

11 On the diagrams on the resource sheet shade in the fraction given next to each diagram.

a b c

12 There are 29 students in a class. 13 of the students are girls.What fraction of the class are girls?

13 In a family of five people, two people are left handed. What fraction of the family are

a left handed b not left handed?

14 There are two red, three blue and six black beads in a box.What fraction of the beads are

a blue b black?

15 There are 75 cars in a car park. 32 of the cars are white.Write down the fraction of the cars that are

a white b not white.

16 Lesley has one geography, three history and five science books.What fraction of her books are

a geography books b geography or history books?

17 Amy says that �13� of this flag is white.

Is Amy correct? Give a reason for your answer.

34

38

15


148

CHAPTER 8 Fractions and decimals

18 Lee chooses some tiles. He wants �35� of each tile he chooses to be blue.

Which of these tiles could Lee choose?A B C D E

8.2 Equivalent fractionsEquivalent fractions are fractions that are equal.

These rectangles areall the same size.

One half of each rectangle is shaded.

The diagrams show that

�12� is equal to �24� and �48�

�12�, �24� and �48� are equivalent fractions.

To find an equivalent fraction, multiply the numerator and the denominator by the samenumber.

Use the diagrams to write down

a fraction that is equivalent to �34�.

Solution 3Shade the same area of the second circle as is shaded in the first. 6 parts will be shaded.

This shows that

�68� is equivalent to �34�

34

68�

�2

�2

Example 3

12

24

48

12

24�

�2

�2

12

48�

�4

�4


149

8.2 Equivalent fractions CHAPTER 8

Complete �23� = �12�

Solution 4

Shade �13� of the circle.

Solution 5The circle has six equal parts so change �13� to sixths.

�13� is the same as �26� so shade two of the six parts of the circle.

Exercise 8B

In questions 1 to 15, copy the fractions and fill in the missing number to make thefractions equivalent.

1 �13� � �9� 2 �15� � �10� 3 �17� � �2� 4 �23� � �9�

5 �34� � �8� 6 �47� � �14� 7 �56� � �15� 8 �35� � �12�

9 �38� � �32� 10 �49� � �45� 11 �78� � �28� 12 �170� � �100�

13 �152� � �15� 14 �2

90� � �60� 15 �1

85� � �40�

In questions 16 to 21 shade the given fraction on the diagrams on the resource sheet.

16 17 18

19 20 21

45

710

58

35

34

16

13

26�

�2

�2

Example 5

23

812�

�4

�4

Example 4

�


8.3 Simplifying fractionsA fraction can be simplified if the numerator and denominator can be divided by the samenumber.This process is called cancelling.When a fraction cannot be simplified, it is in its simplest form or in its lowest terms.

Find the simplest form of the fractions a �150� b �13

80�

Solution 6a Divide both 5 and 10 by 5

�12� cannot be simplified.

The simplest form of �150� is �12�

b Method 1Divide both 18 and 30 by 2

Then divide both 9 and 15 by 3

There is no number that will divide exactly into both 3 and 5 so the

simplest form of �1380� is �35�

Method 26 is the largest number that goes exactly into both 18 and 30 (in other words, 6 is the HCF of 18 and 30)If the HCF is used, then only one step is needed to simplify the fraction.

The simplest form of �1380� is �35�

Prateek has 24 toy cars. 10 of these cars are blue.What fraction of Prateek’s toy cars are blue? Give your fraction in its simplest form.

Solution 7Prateek has 10 blue cars out of 24 toy cars.

�1204� of the toy cars are blue.

Both 10 and 24 are even numbers so divide both numbers by 2

�1204� � �1

52�

�152� cannot be simplified further so �1

52� of the toy cars are blue.

Example 7

Example 6

150


510

12�

�5

�5

915

35�

�3

�3

1830

915�

�2

�2

1830

35�

�6

�6

1024

512�

�2

�2

also assessed in Module 2


151

8.4 Ordering fractions CHAPTER 8

Exercise 8C

In questions 1 to 15 write each fraction in its simplest form.

1 �36� 2 �155� 3 �68� 4 �2

71� 5 �14

00�

6 �2205� 7 �11

26� 8 �12

60� 9 �1

2050� 10 �39

00�

11 �2540� 12 �44

28� 13 �11

05

00� 14 �1

8200� 15 �2

7050�

In questions 16 to 20 write down the fraction of the shape that is shaded.Give each fraction in its simplest form.

16 17 18

19 20

In questions 21 to 25 give each fraction in its simplest form.

21 A class contains 30 students. 12 of these students are girls.Write down the fraction of the class that are girls.

22 Jack has 40 model farm animals. Eight of the animals are horses.Write down the fraction of the model farm animals that are horses.

23 There are four red, three blue and five yellow counters in a bag.Write down the fraction of the counters that are blue.

24 In a car park there are 20 silver, 16 blue, nine red and three green cars.Write down the fraction of the cars that are

a silver b blue c red or green

25 There are 30 cakes in a shop. Five of the cakes are chocolate.Write down the fraction of the cakes that are not chocolate.

8.4 Ordering fractionsHere are two rectangles which are the same size.

The second rectangle has more parts shaded than the first rectangle.

This shows that �170� is bigger than �1

30�

When fractions have the same denominator,you can compare the numerators to put the fractions in order.

�130� of this rectangle is shaded

�170� of this rectangle is shaded


152


Put the fractions �79�, �49�, �89� and �29� in order of size.

Start with the smallest fraction.

Solution 8All the fractions have the same denominator so compare the numerators to put thefractions in order of size.

�29�, �49�, �79�, �89�

Ben shades �34� of a rectangle. Lucy shades �45� of an identical rectangle.

Who has shaded in more of their rectangle? Give a reason for your answer.

Solution 9Compare the fractions by writing them with a common denominator.

The denominators 4 and 5 both divide exactly into 20

Find a fraction equivalent to �34� that has a denominator of 20

Find a fraction equivalent to �45� that has a denominator of 20

�1260� is bigger than �12

50� so �45� is bigger than �34�

As �45� is bigger than �34�, Lucy shaded in more than Ben.

Which fraction is bigger �13� or �25�?

Solution 10The smallest number that the denominators 3 and 5 both divide exactly into is 15

�165� is bigger than �1

55�

So �25� is bigger than �13�

13

515�

�5

�5

25

615�

�3

�3

Example 10

Example 9

Example 8

34

1520�

�5

�5

45

1620�

�4

�4


Write the fractions �14�, �120� and �35� in order of size. Start with the smallest fraction.

Solution 11The smallest number that the denominators 4, 10 and 5 all divide exactly into is 20

Find an equivalent fraction for each of �14�, �120� and �35� with a denominator of 20

Starting with the smallest fraction, the order is �240�, �2

50�, �12

20�

that is �120�, �14�, �35�

Exercise 8D

In questions 1 to 10 write the fractions in order of size. Start with the smallest fraction.

1 �35�, �170� 2 �58�, �34� 3 �34�, �23� 4 �56�, �34�

5 �23�, �56�, �172� 6 �2

90�, �45�, �34� 7 �1

45�, �13�, �1

30�

8 �34�, �196�, �58� 9 �24

30�, �1

70�, �35�, �12

30� 10 �12�, �35�, �1

52�, �13

10�, �1

75�

11 Julie and Susan have identical chocolate bars.

Julie eats �34� of her chocolate bar. Susan eats �78� of her chocolate bar.

Who eats more chocolate? You must give a reason for your answer.

12 Ahmid says that �172� is bigger than �56� because 7 is bigger than 5

Is Ahmid correct? You must give a reason for your answer.

8.5 Improper fractions and mixed numbersAn improper fraction is one in which the numerator is greater than the denominator.

�54�, �

152� and �

184� and are all improper fractions.

The improper fraction �54� can be thought of as ‘5 over 4’, or as ‘5 quarters’.

Similarly, the improper fraction �152� can be thought of as ‘12 over 5’, or as ‘12 fifths’.

A mixed number is a number which has a whole number part and a fractional part.

2�34�, 3�

15� and 6�

78� are mixed numbers.

Mixed numbers can be changed to improper fractions and vice versa.

14

520�

�5

�5

210

420�

�2

�2

35

1220�

�4

�4

Example 11

153

8.5 Improper fractions and mixed numbers CHAPTER 8


To change 2�34� to an improper fraction, work out how many quarters there are in 2�

34�.

There are 4 quarters in 1 and so there are (2 � 4) quarters in 2.Add the extra 3 quarters to get 11 quarters.

So 2�34� � �

141�

Change these mixed numbers to improper fractions.

a 3�12� b 4�

35�

Solution 12a There are 2 halves in 1, so there are 6 halves in 6

Add the extra 1 half to make 7 halves.

3�12� � �

72�

b There are 5 fifths in 1, so there are 4 � 5 � 20 fifths in 4Add the extra 3 fifths to make 23 fifths

4�35� � �

253�

To change an improper fraction to a mixed number, reverse the above process.

To change the improper fraction �167� to a mixed number, firstly work out how many whole

ones there are.

6 sixths is 1; 12 sixths is 2 and so in 17 sixths there are 2 whole ones and 5 sixths.

�167� � 2�

56�

A good way of setting this out is 17 � 6 � 2 remainder 5

2 is the whole number. �56� is the fraction.

Change these improper fractions to mixed numbers.

a �52� b �

145�

Solution 13a 2 halves is 1; 4 halves is 2 and so in 5 halves there are 2 whole ones and 1 half.

�52� � 2�

12�

b 4 quarters is 1; 12 quarters is 3 and so in 15 quarters there are 3 whole ones and 3 quarters.

�145� � 3�

34�

Example 13

Example 12

154


A good way of setting this out is 2�34� ��

(2 � 4

4

) � 3��

1

4

1�


155

8.6 Reading and writing decimals CHAPTER 8

Exercise 8E

1 Change these mixed numbers to improper fractions.a 1�

12� b 1�

13� c 1�

35� d 1�

16� e 1�

78�

2 Change these mixed numbers to improper fractions.a 4�

12� b 2�

14� c 4�

34� d 2�

23� e 4�

25�

3 Change these improper fractions to mixed numbers.a �

75� b �

43� c �

76� d �

181� e �

54�

4 Change these improper fractions to mixed numbers.a �

147� b �

154� c �

257� d �

358� e �

245�

8.6 Reading and writing decimalsThe lengths of two pencils are measured.

The length of the red pencil is exactly 8 cm. This can also be written as 8.0 cm.

The length of the blue pencil is not a whole number of centimetres.

Look at the diagram. Each centimetre is divided into ten equal parts called tenths of acentimetre, also known as millimetres.

A decimal point is used to separate the whole number of centimetres from the number oftenths of a centimetre.The length of the blue pencil is 9.3 cm.

Write down the length of the key.

Solution 14The length of the key is 6 whole centimetres and 8 tenths of a centimetre.The length of the key is 6.8 cm.

Write down the weight of the parcel.

Solution 15The scale measures weight in kilograms.Each kilogram is divided into tenths.The parcel weighs 1.7 kg.

Example 15

0 1 2 3 4 5 6 7 8 9 10 cm

Example 14

0 1 2 3 4 5 6 7 8 9 10 cm

0 1 2 3 4 5 6 7 8 9 10 cm

1 2

30kg


Exercise 8F

In questions 1 to 5 write down the length of each pencil.

1

2

3 4

5

In questions 6 to 8 write down the weight of each parcel.

6 7 8

9 Write down the number that each arrow is pointing to on the scales.

a

b

In questions 10 to 12 write down the weight shown on each scale.

10 11 120

12

3

kg

0

1

2

3

4

5kg

0

kg1

2

3

45

6

7

8

29 30 31 32 33

F G H I

0 1 2 3

A B C D

1

2 3 4

6

5

0kg1

2 3 4

6

5

0kg

1 2

30kg

0 1 2 3 4 5 6cm0 1 2 3 4 5cm

0 1 2 3 4 5 6 7 8 9 10cm

0 1 2 3 4 5 6 7 8 9 10cm

156


0 1 2 3 4 5 6 7 8 9 10cm 11 12 13 14 15

08 Chapter 08 146-166.qxd 1/6/06 11:40 am 6

8.7 Understanding place valueThe decimal point separates the whole number part from the part that is less than 1

Look at the table. The first number on the right of the decimal point tells us how manytenths there are.

There can be more numbers after the decimal point.

The column headings tell us the place value of each figure.

This number in the table is read as ‘four hundred and thirty two point six nine five’.

The column headings tell us that

● the 4 has a value of four hundreds

● the 3 has a value of three tens

● the 2 has a value of two units

● the 6 has a value of six tenths

● the 9 has a value of nine hundredths

● the 5 has a value of five thousandths

Write down the value of the 2 in the number 34.72

Solution 16The 2 has a value of two hundredths.

Write down the number that each arrow is pointing to on the scale.

Solution 17

2.6

2.64 2.75 2.79 2.81

2.7 2.8

2.6 2.7 2.8

Example 17

Example 16

157

8.7 Understanding place value CHAPTER 8

4 3 2 . 6 9 5

thou

sand

shu

ndre

dste

ns

unit

s

tent

hshu

ndre

dths

thou

sand

ths

3 4 . 7 2

tens

unit

s

tent

hshu

ndre

dths


158


Exercise 8G

1 Write down the value of the 6 in each number.

2 Write down the value of the 4 in each number.a 56.43 b 4521.8 c 98.243 d 0.814 e 342.1

3 Write down the value of the 9 in each number.a 3.19 b 792.3 c 0.039 d 79.3 e 1.9

4 Write down the number that each arrow is pointing to on the scales.a b

c d

8.8 Ordering decimals

Five boys took part in a long jump competition.The table shows the distance each boy jumped.

To decide who jumped the furthest,put the distances in order.To put decimals in order,look at the place values.

Use column headings to show the valueof each figure.

Write a 0 in each empty box.

0.4 0.5 0.6

M N P Q

24.5 24.6 24.7 24.8

I J K L

10.1 10.2 10.3

E F G H

6.3 6.4 6.5 6.6

A B C D

a 6 5 2 . 8 1

b 8 7 . 6 3 4

c 1 3 5 . 4 2 6

d 7 5 8 9 . 0 6

e 6 . 5 2

thou

sand

shu

ndre

dste

ns

unit

s

tent

hshu

ndre

dths

thou

sand

ths

Adam 4.3 m

Brian 4.2 m

Colin 4.39 m

Daneep 4.4 m

Elliot 4.33 m

4 . 3 0

4 . 2 0

4 . 3 9

4 . 4 0

4 . 3 3

unit

s

tent

hshu

ndre

dths

All the numbers have a 4 in the units column.

In the tenths column, the smallest number is2 and the largest number is 4

So, 4.2 is the smallest number and 4.4 is thelargest number.


159

8.8 Ordering decimals CHAPTER 8

In order of size, the five numbers are 4.2, 4.3, 4.33, 4.39, 4.4.

Putting the boys’ distances in order gives:

So Daneep jumped the furthest.

Write the numbers 7.53, 7.5, 7.6, 7.65, 7.56 in order of size starting with the biggest.

Solution 18Use column headings to show the value of each figure.Write a 0 in each empty box.As all the numbers have a 7 in the units column, look at the tenths.7.6 and 7.65 both have a 6 in the tenths column but 7.65 has a 5 in the hundredths column so 7.65 is bigger than 7.607.53, 7.50 and 7.56 all have a 5 in the tenths column.To order these numbers use the hundredths column.Starting with the biggest, the order of the numbers is

7.65, 7.6, 7.56, 7.53, 7.5

Exercise 8H

In questions 1 to 12 write the numbers in order of size.Start with the smallest number each time.

1 2

Example 18

4 . 3 0

4 . 3 9

4 . 3 3

unit

s

tent

hshu

ndre

dths

Three of the numbers have a 3 in the tenths column.To order these numbers use the hundredths column.

The numbers in the hundredths column are 0, 9 and 3In order, these are 0, 3, 9

So the three numbers in order are 4.30, 4.33, 4.39.That is 4.3, 4.33, 4.39

Brian 4.2 m

Adam 4.3 m

Elliot 4.33 m

Colin 4.39 m

Daneep 4.4 m

7 . 5 3

7 . 5 0

7 . 6 0

7 . 6 5

7 . 5 6

unit

s

tent

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dths

6 8 . 3 8 3

6 8 . 3 8 7

6 8 . 3 7

tens

unit

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tent

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dths

thou

sand

ths

5 . 7 7

5 . 0 7

5 . 7

unit

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3 6.76, 6.66, 6.67 4 8.11, 8, 8.1, 8.01

5 0.09, 0.9, 0.92, 0.2 6 73.24, 73.2, 73.42, 73.4

7 2.314, 2.413, 2.134, 2.341, 2.431 8 0.373, 0.37, 0.73, 0.333, 0.733

9 15.8, 15.38, 15.3, 15.833, 15.803 10 0.045, 0.05, 0.0545, 0.055, 0.0454

11 6.067, 6.006, 6.07, 6.06, 6.077, 6.076 12 8.092, 8.9, 8.02, 8.09, 8.2, 8.29, 8.92

8.9 Converting decimals to fractionsUsing place values, some decimals can be converted to fractions

0.7 � �170� 0.06 � �1

600�

0.76 � �170� � �1

600�

� �17000� � �1

600�

� �17060�

To convert decimals to fractions, use the place values of the figures.

Write 0.13 as a fraction.

Solution 19

0.13 � �11030�

The heading of the last column with a figure in it gives the denominator.

Example 19

14 The table shows the time, in seconds,in which five runners ran 100 mWrite down the order in which therunners finished the race.

13 The table shows the heights, in metres,of five children.Write down the children in order of height.Start with the tallest child.

160


Linford 10.2

Dwain 10.02

Roger 10.23

Steve 10.12

Maurice 10.21

Linda 1.34m

Anthony 1.4m

Chris 1.43m

Ian 1.33m

Julie 1.3m

0 . 1 3

unit

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ths


Write 0.024 as a fraction.Give your fraction in its simplest form.

Solution 20

0.024 � �120

400�

The heading of the last column with a figure in it is thousandths, so the denominator is 1000

�120

400� � �5

1020� � �2

650� � �1

325�

Write 3.7 as a fraction.

Solution 21

3.7 � 3�170�

The 3 is the whole number part, the .7 is �170�

Exercise 8I

1 Write the decimalsas fractions

In questions 2 to 15 write each of the decimals as a fraction in its simplest form.

2 0.7 3 0.14 4 0.123 5 0.08 6 0.093

7 0.006 8 0.72 9 0.2 10 0.242 11 2.5

12 25.06 13 12.8 14 6.17 15 2.84

Example 21

Example 20

161

8.9 Converting decimals to fractions CHAPTER 8

0 . 0 2 4

unit

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thou

sand

ths

3 . 7un

its

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thou

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a 0 . 3

b 0 . 0 7

c 0 . 1 9

d 0 . 2 5 3

e 0 . 0 8 9

unit

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8.10 Converting fractions to decimalsAll fractions can be written as decimals.

The fractions and decimals in the table are ones that are used frequently and should be learnt.

Other fractions can be changed into decimals.

Write the following fractions as decimals

a �190� b �1

2030�

Solution 22

a �190� � 0.9 b �1

2030� � 0.23

Write the following fractions as decimals. a �25� b �1215�

Solution 23Method 1 – using equivalent fractions

a �25� � �140� � 0.4 b �12

15� � �1

4040� � 0.44

Method 2 – using a calculator

a �25� means 2 � 5 b �1215� means 11 � 25

Using a calculator, Using a calculator,

2 5 0.4 11 25 0.44

�25� � 0.4 �1215�

� 0.44

Short division is suitable for changing �25� to a decimal because the denominator issmall.

�25� means 2 � 5

�25� � 0.4

Not all fractions can be written as exact decimals.

�13� � 1 � 3 � 0.33333….

In this decimal, the 3 keeps repeating.

0 . 45 �2 .20

��

Example 23

Example 22

162


also assessed in Module 4

Decimal Fraction

0.01 �1100�

0.1 �110�

0.25 �14�

0.5 �12�

0.75 �34�

2.0 is the same as 2 so divide 2.0 by 5

5 does not divide into 2 so put down a zero and carry 2

5 divides into 20 four times

�

�


When a decimal has repeating figures, it is called a recurring decimal.

To show that a figure recurs, put a dot above the figure.

So 0.33333… is written as 0.3.

and �13� � 0.3.

Sometimes, more than one figure recurs,

�131� � 3 � 11 � 0.272727….

Put a dot above each recurring figure.

So �131� � 0.2

.7.

Write the following fractions as decimals

a �79� b �1232� c �57�

Solution 24

a �79� means 7 � 9

Using a calculator, 7 9

0.777777… � 0.7.

b �1232� means 13 � 22


0.5909090… � 0.59.0.

c �57� means 5 � 7


0.714285714… � 0.7.14285

.

Exercise 8J

1 Write the following fractions as decimals.

a �190� b �1

3070� c �1

300� d �1

506010� e �10

800�

Example 24

163

8.10 Converting fractions to decimals CHAPTER 8

Work out 7 � 9 on a calculator.

The 7 recurs so put a dot above the 7


The 90 recurs so put a dot above each of these figures.

Do not put a dot above the 5, as it does not recur.


A group of six figures recurs.There isn’t enough room to see all the figures recurring butyou can see that the same pattern of figures is starting again.

When more than two figures recur, just two dots are used,one above the first figure in the recurring group and oneabove the last figure in the group.

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�

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2 Write the following as equivalent fractions and then as decimals.

a �45� � �10� b �570� � �100� c �2

85� � �100� d �5

900� � �1000� e �2

30� � �100�

3 Write down the following fractions as decimals.

a �12� b �14� c �110� d �1

100� e �34�

4 Use short division to change these fractions to decimals.

a �35� b �38�

5 Use a calculator to change these fractions to decimals.

a �18� b �490� c �22

35� d �78� e �11

16�

6 Use a calculator to change these fractions to decimals.

a �23� b �89� c �151� d �1

72� e �17�

Chapter summary

164


�

You should know and be able to use these facts

The top number of a fraction is called the numerator.

The bottom number of a fraction is called the denominator.

Equivalent fractions are fractions that are equal.

A fraction can be simplified if the numerator and denominator can both be dividedby the same number. This process is called cancelling.

A fraction that cannot be simplified is in its simplest form.

To compare fractions, first write them with the same denominator.

In a decimal the decimal point separates the whole number part from the part that isless than one.

Decimals can be put in order by looking at the place value of each number. Firstcompare the whole number part, then the tenths, then the hundredths, then thethousandths.

Decimals can be converted to fractions by using their place value.

Fractions can be converted to decimals by using equivalent fractions or division.

Some fractions convert to recurring decimals.

change an improper fraction to a mixed number and vice versa

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165

Chapter 8 review questions CHAPTER 8

Chapter 8 review questions1 Write down the fraction of each shape that is shaded.

Give each fraction in its simplest form.a b c d

2 Write down the fraction of this shape that is shaded.Give your fraction in its simplest form.

3 a Write down the fraction of this shape that is shaded.Write your fraction in its simplest form.

b Shade �23� of this shape

on the resource sheet.

(1387 June 2003)

4 There are 60 cars in a car park. 35 of the cars are silver.Write down the fraction of cars in the car park that are silver.Give your fraction in its simplest possible form.

5 Copy the fractions and fill in the missing number to make a pair of equivalent fractions.

a �45� � �15� b �38� � �6� c �190� � �50� d �58� � �20�

6 a Shade �14� of this shape.

b Copy the fractions and write a number on the dotted line so that the two fractions are equivalent

�14� � �1…2�

7 Give each fraction in its simplest form.

a �48� b �155� c �14

00� d �1

7050�

8 Write down the reading on each of these scales.

a b c

9 Write down the value of the 6 in each of the following numbers.a 56.3 b 9.62 c 0.916 d 45.16

1

20 kg

1

20 kg

A B C

3.6 3.7 3.8

�


10 Five girls each threw a ball in a competition.The table shows the distance, in metres,each girl threw the ball.Write down the distances in order of size.Start with the longest distance.

11 Write each fraction as a decimal. You may not use your calculator.

a �170� b �1

900� c �1

4030� d �1

60

700� e �25� f �2

70�

12 Write 0.45 as a fraction. Give your fraction in its simplest form.

13 Write 0.028 as a fraction. Give your fraction in its simplest form.

14 Here are two fractions �35� and �23�

Explain which is the larger fraction.You may use the grids to help with your explanation.

(1387 June 2003)

15 Change �78� to a decimal.

16 Amanda and Mary each had the same size of chocolate bar.

Amanda ate �23� of her bar of chocolate. Mary ate �58� of her bar of chocolate. Work out

which girl had eaten the most chocolate. You must give a reason for your answer.

17 Write these five numbers in order of size. Start with the smallest number

2.5, 0.5, 0.52, 2.2, 0.25 (1388 Jan 2003)

18 �172�, �56�, �23�

Write these fractions in order of size. Start with the smallest fraction.(1388 Mar 2002)

19 Write these five fractions in order of size. Start with the smallest fraction.

�25�, �13�, �12�, �38�, �141�

20 a Write 0.35 as a fraction. Give your answer in its simplest form.

b Write �38� as a decimal. (1387 June 2002)

21 Use your calculator to write each fraction as a decimal.

a �2430� b �56� c �11

16� d �1

41� e �9

70�

22 Change to improper fractions.

a 2�23� b 8�

45�

166


Anna 20.4

Bianca 19.96

Chaya 19.9

Debbie 20.34

Eloise 20.04

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00 found-prelims ii-vi - pearson education · 38 pythagoras’ theorem 573 38.1 pythagoras’...

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