9 Fractions

C H A P T E R

9Fractions

What you will learn9.1 Naming fractions

9.2 Equivalent fractions

9.3 Comparing fractions

9.4 Adding fractions

9.5 Subtracting fractions

9.6 Multiplying fractions

9.7 Dividing fractions

9.8 Percentages

9.9 Operations with percentages

9.10 Ratios and fractions

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VELS NumberWrite equivalent fractions for a fractiongiven in simplest form (for example,

). Know the decimal

equivalents for the unit fractions

, and find equivalent representations of

fractions as decimals, ratios and percentages.

Understand ratio as both set : set comparisonand subset : set comparison, and find integerproportions of these, including percentages.

Write the reciprocal of any fraction andcalculate the decimal equivalent to a givendegree of accuracy.

Use efficient mental and/or written methodsfor arithmetic computation involving rationalnumbers, including division of integers by two-digit divisors.

Use technology for arithmetic computationsinvolving several operations on rationalnumbers of any size.

19

18

15

,

14

,13

,12

,

23

46

69

. . .

Breaking the recordIn the modern world there seems to bean increasing need for greateraccuracy in measuring length, time,weight and other things. Thisincreasing need for accuracy hasmeant that we often need to usesmaller parts of the unit formeasurement being used. These partsare the fractions that we will study inthis chapter. For example, worldrecords can now be broken by animprovement of a mere one-hundredthof a second.

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Essential Mathematics VELS Edition Year 7

Do now

294

Skillsheet

T EACHE R

1 List the prime factors of the following:a 24b 70c 32d 41

2 What are the highest common factors of each of the following?a 12 and 30b 16 and 48c 12, 20 and 44

3 Which of the following represent ?

a b c

4 If each yellow shape is , what is the fraction represented by the orange region?

a b c

5 Solve the following:

a

b

c

d 2 lots of

6 Arrange the following into ascending order: 2, 1.6, 0.5, 2.01,

Answers

1 a 2, 3 b 2, 5, 7 c 2 d 41 2 a 6 b 16 c 4 3 a and c 4 a b c 5 a b c d 1

6 0.5, 1.6, 2, 2.013237

,34

,

12

23

134

34

13

23

16

3237

34

,

34

3 213

112

14

12

14

13

13

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Chapter 9 Fractions 295

Key ideas

is a common or proper fraction

as it is less than 1.

is an improper fraction as it is

greater than 1 and is called a

mixed number.

is a mixed number as it consists of a fraction and a whole number.214

94

44

44

14

214

214

94

38

9.1 Naming fractionsFractions are used in everyday life to describe parts of quantities. Each of the following is

said to be three-eights of the whole and is written as :

Although the diagram on the right shows three of the eight parts

shaded we do not write this as as the parts are not equal.38

1000 mL

500 mL375 mL

Volume in amarked measuring

cylinder

38

Example 1

For each of the following:

a What fraction of the diagrams are shaded yellow?b Write the fraction in words.

i ii

numerator

denominator

number of parts out of the whole

line (vinculum)

number of equal parts all together

38

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Example 2

Write the improper fraction as a mixed number.154

ExplanationSolution

154

334

154

44

44

44

34

Example 3

Rewrite the mixed number as an improper fraction.223

Solution

223

83

223

2 3 2

3

83

1Example 1 For each of the following:

i What fraction of the diagrams are shaded?ii Write these fractions in words.a b c d

9AExercise

ExplanationSolution

a i

ii2

27

512

5 Parts shaded

12 Total number of parts

2 Shapes shaded

7 Total number of shapes

b i Five-twelfths ii Two-sevenths

The names tell the number of parts ofinterest and the total number of parts.

Explanation

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2Example

3Example

e f g

h i

2 Write each of the following as a proper fraction:

a one-half b three-quarters c three-fifthsd five-twelfths e four-sevenths f six-tenthsg four-sevenths h two-thirds i five-eighths

3 Write these improper fractions as mixed numbers:

a b c d e f g h

4 Illustrate these mixed fractions using rectangles and shade the fraction given:

a b c d e f g h

5 Rewrite each of the following mixed numbers as improper fractions:

a b c d e f

6 Write whether each of the following is a mixed number, a proper fraction, an improperfraction or a whole number:

a b c d e f g h

7 Using a diagram show how large cakes could be equally shared amongst

eight people at a sleep-over.

214

205

1020

1312

79

75

55

235

415

416

28

113

49

127

213

135

137

512

214

124

235

314

138

213

113

2020

1510

114

52

95

74

32

Enrichment: Egyptian fractions

8 The ancient Egyptians only had a way of writing uniaryfractionsone-fifth, one-seventh, one-everything. So

to represent they needed to write it as or

in shortest form as

Think of the family of twelfths. They would have

written as or as .

How would they have written , , . . . , ?1212

412

312

16

112

112

212

12

14

.

14

14

14

34


Th

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9.2 Equivalent fractionsEquivalent fractions have the same value.

This rectangle has been divided into a number of different equal parts.

The shaded area of the rectangle is the same in each case and so these fractions are equal.We call these equal fractions equivalent fractions.

36

24

12

Key ideas

An equivalent fraction can be created by:

multiplying the numerator and denominator by the same number. is the same as

dividing the numerator and the denominator by the same number. This is called

ccaanncceelllliinngg::

A fraction that cannot be cancelled is said to be a ssiimmppllee fraction.

63

105

35

3 25 2

6

1035

Example 4

ExplanationSolution

Fill in the missing number to make an equivalent fractions: 25

8

20

25

2 45 4

8

20To get the numerator of 8 requires multiplyingby 4. If it is to be an equivalent fraction we mustalso multiply the denominator by 4.

Example 5

Write three fractions that are equivalent to .25

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Example 6

ExplanationSolution

Write the fraction in its simplest form.1216

123

164

34

Divide numerator and denominator by 4.

4Example

5Example

9BExercise

1 Fill in the missing numbers to make an equivalent fractions each time:

a b c d

e f g h

2 Write four fractions that are equivalent to each of the following fractions:

a b c d e f

g h i j k l

3 Fill in the missing numbers to complete the sets of equivalent fractions each time:

a b

c d

4 Write each of the following fractions in its simplest form:

a b c d e f

g h i j k l30

1002080

1230

1421

184

2440

1236

1510

86

1030

414

510

56

10

18

30

50

90073

70

39

3000

77

3300

14

3

16

10

44

40025

4

15

8

10

30

512

47

98

35

49

611

310

56

27

34

25

13

73

927

4

53

934

8

3618

12

2418

31632

4

810

4

6Example

ExplanationSolution

25

4

10

615

8

20

25

44

8

2025

33

6

1525

22

4

1025

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Count by 2s 2

Count by 5s 5

Count by? 2 6 14

Count by? 3 9

Count by 1s 1 2 3 4 5 6 7 8

Count by 3s 3 6 9 12 15 18 21 24

5 Fill in a fraction wall. Part of it has been completed to show you the idea. Ask yourteacher for a copy.

On your fraction wall show the sets of equivalent fractions for:

a b

Keep your fraction wall for future reference.

6 Look at the pattern in this table. If we form each pair into a fraction we get the following series of equivalent fractions:

, , , . . . ,

a Using your fraction wall or otherwise, explain why this works.b Copy and complete these tables:

i

ii

824

39

26

13

57

13

1 2 4 16 30

Enrichment: Fair share

7 Show how each of the following can be divided into the number of equal areas required:

a four equal areasof the same shape

b three equal areasof the same shape

c two equal areas thatare different in shape

d A way of cutting a circular birthday cake into equal slices is todivide the circumference into equal lengths and cut to themiddle. Here is an example of a cake divided into sevenths forthe birthday girl and her six guests.

Using this method of equally sharing the length around the circumference, try cuttingeach of the following cakes into six equal slices:

Th

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9.3 Comparing fractionsEquivalent fractions are useful when comparing fractions that have different denominators.

It is very hard to decide if or is a bigger fraction.911

45

Key ideas

To compare fractions we need to convert each of them into equivalent fractions that havethe same denominator.Fractions with the same denominator are said to have ccoommmmoonn ddeennoommiinnaattoorrss. Thesmallest common denominator is the most appropriate to use and we call this the lloowweessttccoommmmoonn ddeennoommiinnaattoorr or LLCCDD.

Example 7

ExplanationSolution

Determine which is the larger, or , and state the LCD.712

58

LCD (8, 12) 24

5 3

7 2

is larger than 712

58

1424

212

712

1524

38

58

Multiples of 8: 8, 16, 24, 32, . . .

Multiples of 12: 12, 24, 36, . . .

Find an equivalent form of both fractionswith the LCD and compare.

Because is larger than 1424

1524

Example 8

Arrange the following from largest to smallest: 5 , 173

310

514

,

ExplanationSolution

LCD (4, 10, 3) 60

21 15 30560

154

214

5310

5 10 3

10

5310

and 173

5 14

5 4 1

4

214

Convert all fractions to mixed numbers.

Find LCD.

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9CExercise

7Example

8Example

1 Determine which of the fractions is the larger fraction, and state the LCD:

a b c d e f

2 Arrange each of these sets of fractions in descending order (from largest to smallest):

a b c

3 Write each of the following sets of fractions in ascending order (from smallest to largest):

a b c

d 1 e f

4 Rewrite the fractions in each set with their lowest common denominator before youfind the next fraction in the pattern each time:

a b c

d e f12

, 47

, 914

, p79

, 23

, 59

, p56

, 23

, 12

, p

18

, 14

, 38

, p16

, 13

, 12

, p14

, 12

, 34

, p

159

, 102

, 323

, 2118

, 256

323

, 237

, 237

, 6521

, 1010

173

, 2, 315

, 1512

34

,

25

, 134

, 2110

, 78

, 32

25

, 35

, 47

, 38

, 112

23

, 114

, 223

, 78

, 329

616

, 1248

, 1032

, 824

37

, 1221

, 57

, 4249

59

, 23

, 79

, 218

23

, 712

56

, 78

34

, 78

35

, 58

58

, 43

35

, 23

53 6

17 20

Largest to smallest is , 514

5310

,173

34060

203

173

31860

610

5310

Convert each to an equivalent fractionwith the same denominator and comparethem.

Write the answer to the original question.

Enrichment: Make that one half

5 Draw a large triangle of your choice and mark a point inside and call it O.

a Draw lines from O through the three vertices and findthe point on each line halfway to each vertex. Join these midpoints to form a new triangle.

b How do the sides of the new triangle compare to the old?c What results from choosing points that are two-thirds rather than halfway to the

vertices? How do the triangles compare now? Can you think of a rule to describeyour findings? What would have happened if your point O was on the triangle rather than inside? What if it were outside?

Th

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9.4 Adding fractionsFor fractions with the same denominator, add their numerators:

Fractions with different denominators are most easily added by converting each of theminto equivalent fractions with common denomonators.

111

2

11

1 211

311

Key ideas

To add fractions:11 Find a common denominator, usually the lowest common denominator LCD or LCM.22 Convert each of the fractions into their equivalent fraction with the LCD.33 Now add or subtract the numerators.44 If possible, write the answer in simplest form or as a mixed number.

Example 9

ExplanationSolution

Write each of the following as a single fraction:

a b23

16

15

25

a

b23

16

46

16

4 1

6

56

15

25

1 2

5

35

These already have a common denominator,so add the numerators.

Rewrite with an LCD of 6.Add like fractions.

Example 10

ExplanationSolution

Simplify 223

112

.

416

256

16 9

6

166

96

223

112

83

32

Write mixed numbers as improper fractions.

Rewrite fractions as equivalent fractions withthe LCM of 6.Add like fractions.

Write the improper fraction as a mixed number.

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9DExercise

1 Write each of the following as a single fraction or whole number:

a b c d

e f g h

i j k l

m n o

2 Write the fractions with common denominators, and perform the operations:

a b c d

e f g h

3 Simplify:

a b c

d e f

4 Simplify:

a b c

d e f

5 Simplify:

a b 13

10 1

25

2756

112

2012

123

56

3014

523

523

378

125

19

10 3

12

113

249

112

113

112

214

511

11

113

23

113

129

159

247

237

235

115

112

112

59

76

25

57

14

37

25

12

13

34

34

15

14

38

12

14

17

97

27

18

58

38

1110

3

10

510

2381

9

811110

9

1027

27

511

3

11

59

79

35

45

617

6

175

13

813

23

13

49

19

27

37

15

35

9aExample

9bExample

10Example

Calculator keystrokes

13

225

1+

1115

3

+

=

ba

a

a a2

2

3

Answer:

11 15

2 51

1 3

1 3/c b/c b/c

b/c ba /c ba /c

Enrichment: Magic fractions

6 Complete these magic squares:

14

1

12

2 1

23

53

13

3

1

Th

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In the main, subtracting fractions operates in the same way as for addition. The only difference

lies with questions of the type where we need to convert to mixed numbers first.212

123

,

9.5 Subtracting fractions

Example 11

ExplanationSolution

Write each of the following as a single fraction:

a b89

12

79

49

a

b

718

18 9

18

89

12

1618

9

18

13

31

93

39

79

49

7 4

9We already have a common denominator, sowe just subtract.

We can cancel to a simpler fraction.

Write with an LCD of 18.

Subtract like fractions.

Key idea

To subtract fractions, follow the same steps as for addition.

Example 12

Simplify 212

123

ExplanationSolution

59

159

109

212

123

52

53

Convert to improper fractions.

Write fractions as equivalent fractions withLCM 9 and subtract.

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9EExercise


a b c d

e f g h


a b c d

e f g h

3 Simplify:

a b c d

e f g h

i j k l

m n o p

4 Simplify:

a b

c d

5 Scuba divers are aware of the time spent on a dive. Find the total time spent if Peta

spends minutes searching for coral and minutes collecting starfish.

6 This quadrilateral has a perimeter of metres. If the longest

side if metres and is twice the length of the opposite

parallel side, how long is each of the other sides if they are equal?

4412

12834

814

1512

256

116

112

113

113

249

a129

149b

256

116

a112

113b37

8 1

25

37

10 2

310

6123

156

412

13 7

1526

32

11 1

722

729

356

513

414

623

112

425

17

102

13

112

4213

4123

307

10 29

910

225

145

418

158

91013

12

132

56

116

235

125

3511

11

11

410

2

1545

13

1112

34

78

24

59

13

58

14

45

3

1034

23

59

39

29

710

1

10

310

38

28

617

6

17

27

27

511

3

1178

38

56

36

11aExample

11bExample

12Example

Enrichment: Birthday puzzle

7 Today Sallys age is one-seventh of Wans age. Wan is a teenager.

a If in 1 years time she will be one-fifth of his age, what fraction of his age willshe be 2 years from now?

b How many years will need to pass until she is one-third of his age?Th


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Chapter 9 Fractions

What about fractions of fractions? Find

To find of you could think of the as in

the diagram on the right. of this gives:

So or

So the same pattern of multiplying the numerators andmultiplying the denominators also works in this case.

13

25

1 23 5

2

1513

of 25

2

15

13

25

25

13

13

of 25

307

9.6 Multiplying fractions

Key ideas

When multiplying fractions, multiply the denominators and multiply the numerators.

A whole number can be written as afraction with a denominator of 1.

Cancel before multiplying to makethe calculation easier.

Simplify the answer where possible.

Of, times and lots are other namesfor multiplication

3

3 3 2

2 10

1 5

1

of 40 401

34

34

34

74

148

310

32

96

910

26

63

21

23

31

23

Example 13

ExplanationSolution

Find each of the following amounts:

a of 24 b of 4937

14

a of 6

b of 49 21491

37

37

24 14

241

14

means divide into 4 groups.

of 49 is 7, so must be three times this amount.37

17

14

2 equal parts out of 15

2 equal parts out of 5

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Example 14

ExplanationSolution

Simplify: a b c310

54

29

34

27

23

45

a

b

c

1

12

31

102

51

42

21

93

1 1 12 2 3

3

14

342

21

7

3 12 7

8

15

23

45

2 43 5

Multiply the numerators and denominators.

The fraction is in simplest form.

Cancel, then multiply the numerators and thedenominators.The resulting fraction is in simplest form.

Cancel before multiplying numerators anddenominators.

The resulting fraction is in simplest form.

Example 15

ExplanationSolution

Simplify each of the following, giving your answers as mixed numbers:

a b 225

137

112

34

a

b

337

247

225

137

1251

102

7

118

98

112

34

32

34

Convert mixed numbers into improper fractions.

Multiply numerators and denominators.

Write the answer as a mixed number.

Convert into improper fractions and cancel common factors.Multiply new numerators and new denominators.

Write the answer as a mixed number.

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9FExercise

1 Find each of the following amounts:

a of 12 b of 25 c of 20 d of 16

e of 81 f of 30 g of 45 cents h of 65 cents

i of $100 j of $420 k of $200 l of $720

2 Find each of the following amounts:

a of 6 b of 100 c of 35 d of 12

e of 36 f of 21 g of 81 cents h of 88 cents

i of $120 j of $30 k of $480 l of $630

3 A photographer is hired for a wedding. She takes 240 photographs in the 3 hours sheworks.

a of the photographs are taken at the church. How

many of the photographs are taken at the church?

b of the photographers time is spent at the church.

How many minutes is this?

c Of all the photographs, include the bride. How

many photographs are there with the bride included?d How many photographs do not include the bride?

e If the reception takes up of the photographers

time, how much of her time is spent at the reception?

(Give your answer in minutes.)

4 Simplify:

a b c d e

f of g of h i of 79

58

15

49

12

57

27

34

27

of 34

23

of 45

15

19

14

17

16

13

310

45

38

23

57

310

56

34

38

39

37

23

34

45

710

23

18

110

16

14

15

15

12

19

18

14

15

13

13Example

14aExample

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5 Simplify each of the following. (Cancel where possible.)

a b c d

e f g h

6 a b c

d e f

g h i

7 Simplify each of the following, giving your answers as mixed numbers whereappropriate:

a b c d

e f g h

i j k l

8 I bought 4 kg of mixed nuts. of them were peanuts. How many kilograms of peanuts

did I buy?

9 Christopher needs cans of paint for his bedroom. The area to be painted in

Matthews room is times the area in Christophers room. How much paint will

Matthew require?

129

213

38

227

11

204

111

318

225

of 314

156

212

238

123

112

234

34

of 149

227

37

23

135

149

53 235

113

of 5

421

35

78

58

14

8

1556

23

9

10

1235

2122

1120

1227

58

9

109

20

1021

1

12

15

38

79

23

45

67

12

35

7

11

725

1021

725

58

37

of 49

518

9

11

23

35

57

15

13

of 38

56

1514bExample

14cExample

15Example

Enrichment: River flow

10 This diagram shows the fraction of flow in the channels of a river. The fraction of theflow at the start of each channel is shown. What fraction of the full river flow wouldpass A, B, C and D?

A

D

C

B

25

35

23

13

35

12

12

14

Th

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From the last section we discovered that multiplying a quantity by was the same as

dividing by and 240 2 120. This suggests a rule:

To calculate 240 n, solve 240 1/n. If you try a few values for n you will see that it isalways true.

2: 240 12

120

12

9.7 Dividing fractions

Key ideas

If you want to divide by a fraction, multiply by the reciprocal of the fraction.

The reciprocal of the fraction is the fraction .

The product of a number and its reciprocal is always equal to 1:

1

If you wanted to divide by . . . multiply by the reciprocal

b

a

a

b

32

23

3 13

12

2

b

a

a

b

b

a

a

b

Example 16

ExplanationSolution

Write the reciprocal for the each of the following:

a b c 5 d 123

13

45

a

b or 3

c

d35

15

31

54

The reciprocal is found by inverting the fraction.

Invert the fraction.

Think of 5 as and then invert.

Convert to an improper fraction and then invert.

51

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Example 18

Simplify each of the following:

a b 5 13

4 45

1420

7

15

Example 17

Simplify each of the following:

a b 4 23

12

5

ExplanationSolution

a

b

6

61

4 23

42

1

321

1

10

12

5 12

15

Change division to multiply by the reciprocal.Multiply the numerator and denominator.

Change division to multiply by the reciprocaland cancel before multiplying numerators anddenominators.

ExplanationSolution

a

b

119

109

2 53 3

162

3

5243

5 13

4 45

163

245

112

32

21 342 1

1420

7

15

142

20

1571

Change division to multiply by the reciprocal andcancel before multiplying numerators anddenominators.

Express the improper fraction as a mixed number.

Rewrite the mixed numbers as improper fractions.Now change division to multiply by the reciprocaland cancel by common factors.

Now multiply numerators and denominators.

Finally, write as a mixed number.

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9GExercise

1 Write the reciprocal of each of the following:

a b c d e f

g h i 2 j 3 k 7 l 11

m n o p

2 Simplify each of the following:

a b c d

e f g h

i j k l


a b c

d e f

g h i

j k l

m n o


a b c

d e f

g h i

j k l 178

212

234

317

212

178

712

318

315

227

234

127

25

33

1027

114

34

215

125

34

113

12

112

13

1450

21

10034

6

11512

16

59

1011

915

1825

320

9

10

512

56

211

34

18

59

25

37

56

67

17

57

13

23

12

13

12

12

310

1538

74 23

3 45

2 13

9 16

7 15

5 14

111

218

815

613

4

437

11120

238

114

178

19

15

18

519

417

37

23

16Example

17Example

18Example

Calculator keystrokes

13

225

1

231

+

=

2

2

5

Answer:

3

2 51

1 3

1 3a b/c a b/c a b/c

a b/c a b/c a b/c

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5 There are pizzas left over after tea. If six people are to share them for lunch thenext day, what fraction of a pizza will each person get?

6 A farm of 32 hectares is to be split into paddocks each measuring hectares. Howmany paddocks can be made?

7 Ice-cream is scooped out at about of a litre per scoop. If I have litres left, how

many scoops will I be able to serve?

523

215

315

134

8 A grocer has twelve-and-a-half kilograms of mandarins to pack into three-and- a-quarter kilogram bags. How many bags can he fill and how many kilogramsof mandarins will be left?

9 Trays of meat at the butchers shop weigh kg. If Kay bought six trays for her

restaurant, and each meal she prepares for her customers requires kg of meat, how

many meals can Kay prepare?

916

214

Enrichment: Measuring time

10 In days of old, one way of measuring time was to mark a burning candle.Candles burn at a fairly constant rate.

If this type of candle takes 6 hours 20 minutes to be used completely,where will it have burn down to by 12 noon if lit at 9.00 in the morning?

Make an accurate scale drawing of this candle and mark every 20 minutes, then mark where it will have burnt down to by midday.

Research on the web the history of time.

40 cmTh

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9.8 PercentagesAdvertisements involving percentages are used all the time.

Key ideas

The symbol %% means ppeerr cceenntt. This comes from the Latin words per centum and means

oouutt ooff 110000. So 7% means 7 out of 100. We can write this as or 0.07.

Decimal shortcut: When dividing by 100 move the decimal point two places to the left:

15% 0.15

124% 1.24

12.5% 0.125Some percentages are used so often that it helps to remember their fraction equivalent:

7100

Example 19

Express each of these percentages as a fraction or mixed number in its simplest form:

a 13% b 80% c 125% d 6623

%

5% 10% 20% 25% 33.3% 50% 66.6% 75% 100% 130%

0.05 0.1 0.2 0.25 0.33. . . 0.5 0.66... 0.75 1 1.3

1310

11

34

23

12

13

14

15

110

120

ExplanationSolution

a

b

45

80% 80 4

1005

13% 13

100Write as a fraction of 100.

Write as a fraction of 100 and cancel.

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c

d

23

2002

3003

2003

1

100

200

3

1001

6623

% 2003

%

114

54

125% 125 5

1004Write as a fraction of 100; cancel by the commonfactor of 25.

Write as a mixed number.

Convert the mixed number percentage into animproper fraction percentage.Convert the percentage to a fraction.

Use the reciprical.

Cancel by the common factor of 100.

Example 20

ExplanationSolution

Express these percentages as decimals:

a 15% b 124% c 12.5%

a

Or

b

Or

c

Or 12.5% 0.125 0.125

125

1000

12510

%

12.5% 12510

%

124% 1.24 1.24

124

100

124% 124100

15% 0.15 0.15

15% 15

100Write the percentage as a fraction.

Convert the fraction to a decimal.Or move the decimal point two places to the left.

Write the percentage as a fraction.

Write the fraction as a mixed number.

Convert the fraction part to a decimal.Or move the decimal point two places to the left.

Rewrite as a fraction.

Write the percentage as a fraction.

Convert the fraction to a decimal.Or move the decimal point two places to the left.

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Example 21

Convert each of the following to a percentage:

a 0.26 b 0.567 c d e

Hint: Decimal shortcut:1 Convert the fraction to a decimal:

2 Move the decimal point two places to the right, i.e. multiply by 100.

56

5 6 0.8333

123

38

910

ExplanationSolution

a 0.26 26%

b 0.567 56.7%

c or

d or

e or

166.6#%

5003

% 166.6#%

53

5 100

3%1

23

1.666

37.5% 37.5%

3 10025

82%

752

%38

0.375

90% 90%

910

90

1009

10 0.90

Multiplying by 100 so move the decimalpoint two places right.Multiplying by 100 so move the decimalpoint two places right.Multiplying by 100 so move the decimalpoint two places right, which requiresinserting a zero.Multiplying by 100 moves the decimal point two places right.

Multiplying by 100 moves the decimal point two places right.

9HExercise

19Example

20Example

1 Express each of these percentages as a fraction or a mixed number in its simplestform:

a 19% b 53% c 71% d 29% e 70%f 25% g 10% h 50% i 200% j 130%k 145% l 170% m 360% n 420% o 301%

p 208% q r s t

2 Express these percentages as decimals:

a 35% b 27% c 50% d 47% e 7% f 1%g 9% h 5% i 132% j 256% k 145% l 260%m 12.2% n 24.5% o 99.9% p 16.3%

338

%1012

%1215

%1314

%

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3 Convert each of the following to a percentage:

a 0.34 b 0.57 c 0.26 d 0.06

e 0.456 f 0.214 g h 0.716i 1.67 j 2.456 k 1.575 lm 0.012 n o 3 p 0.004

4 Convert each of the following to a percentage:

a b c d e

f g h i j

k l m n o

p q r s t 118

156

2310

134

311

59

38

13

1780

116

1140

1275

2750

710

34

25

99100

74100

5100

3100

0.3# 2.66

#0.33#

5 Copy and complete this table:

Enrichment: Learning how to make money

6 This is a tangram, and each of the parts are called tans. Working in a group as directed by your teacher, cut out each of the tans and write on them their percentage of the original tangram.

Fit the tans together to make these shapes, make a sketch of each in your workbook and record the percentage of the original tangram.

How many different percentages can you make using one, two or more of the tan pieces?

Percentage Fraction Decimal

0%0.1

25%

0.37550%

62.5%

0.75

23

13

18

21Example

Th

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You should now be able to change between fractions, decimals and percentages, andperform operations on them. This skill allows you to make a decision as to whether, forexample, a discount of 37% or one-third off is the best option.

9.9 Operations with percentages

Key ideas

so 100% is one whole.

If the large square on the right represents one whole, and 40% isshaded, then the unshaded region is 60% because 100% 40% 60%Calculations with percentages are of two types:11 We can find a percentage of a quantity.22 We can express one quantity as a percentage of another and

we can find a percentage of a given amount.In both cases we use the decimal or fraction equivalent to make the calculation.

100% 100100

1,

Example 22

ExplanationSolution

Find:

a 100% 70% b 60% 30%

a 100% 70% 30%b 60% 30% 90%

We can add and subtract percentages asthey are like quantities.

Example 23

ExplanationSolution

Find 25% of 40.

25% of 40

10

401

2540

Convert to a fraction calculation.

Example 24

A class of 25 students has 15 girls. What percentage of the class are girls and whatpercentage are boys?

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ExplanationSolution

The fraction of the class who are girls

is

The percentage of the class who are girls

is

The percentage of the class who are boys

is 100% 60% 40%

15251

100 4% 60%

1525

.

Start by expressing the required quantityas a fraction.

Convert to a fraction, then cancelcommon factors to make calculationsimpler.Subtract from the whole ( 100%).

Example 25

ExplanationSolution

Of 50 people surveyed, 8% eat chocolate on a regular basis. What is the number of peopleeat chocolate on a regular basis?

4So four people eat chocolate on a regularbasis.

8% of 50 8

1002 501 Convert the percentage into a fraction

and simplify.

Write the answer in words.

9IExercise

22Example

23Example

24Example

1 Find:

a 100% 30% b 100% 90% c 100% 50%d 100% 8% e 100% 15% f 100% 2%g 60% 30% h 55% 20% i 40% 45%j 5% 9% k 13% 77% l 33% 47%

2 George used 15% of the firewood. What percentage was left?

3 55% of children born on Thursdays are male. What percentage are female?

4 Pauline has collected 80% of a set of football cards. What percentage does she need tocomplete the set?

5 Find:

a 20% of 400 b 30% of 2000 c 15% of $400d 45% of $300 e 75% of 80 f 40% of 110g 64% of 120 h 38% of 740 i 85% of 250j 32% of 180 k 12% of 480 l 18% of 70

6 Kathryn scores 45 marks out of a total of 80. What percentage score does she receive?

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25Example

7 James played 9 games out of 36 for the local squash club. What percentage of thegames did he play?

8 18 players of a 24-member squad voted for Patricia as captain.

a What percentage voted for Patricia?b What percentage did not vote for Patricia?

9 John earns $80 for a week of work. This is made up of $50 for odd jobs, $20 formowing the lawns and the rest for cleaning the car.

a What percentage of the $80 comes from odd jobs?b What percentage comes from car cleaning?

10 56% of a class of 25 students can sing. How many students is this?

11 A shop orders 500 packets of chips per week. If 24% are sold on Monday:

a what percentage is left? b how many packets have been sold?

12 A stereo costs $850. If 20% is required as a deposit to lay-by the set, how muchdeposit must be paid?

13 A store is giving 30% discount on its products. If a dinner set costs $700:

a what will you save on the dinner set? b what is the new price?

14 15% of a Year level play cricket. If the Year level is made up of 40 pupils, how manyplay cricket?

15 A survey of skilled workers found that 57 were over the age of 50 years. If thisrepresented 37% of the work force, how many workers were there altogether?

Enrichment: Grid game

16 Take it in turns to roll a dice, and with each roll enter the score in the top or bottom box on the right to make a fraction.

Colour that fraction on the grid. The winner is the first to mark a path from one side to the other.

1 5 1 11525

1 2 2 4 31212

14

3 1 1 21223

6 1 2 11313

16

1 1223

45

34

56

15

2 1 4535

12

12

12

13

Th

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9.10 Ratios and fractionsWhen we need to compare two quantities, we often use aratio. Consider the situations below.

The ratio of the number of engines to the number ofcarriages in the train is 1 to 3. This is written as 1 : 3. Theratio of the number of carriages to the number of engines is3 to 1, or 3 : 1.

Key ideas

A ratio compares quantities, say a to b, and is written as a : b. If we wanted to compare b to a we would write it as b : a.

Another way to write a ratio is as a fraction. The ratio a : b can be written as a fraction

Ratios, like fractions, have equivalent forms. A ratio that cannot be simplified is said tobe in its simplest form.

a

b.

Example 26

ExplanationSolution

Write each of the following as a ratio and a fraction:

a the number of loaves of bread to the number of bottles of milkb the number of bottles of milk to the number of loaves of bread

a 5 : 3 or

b 3 : 5 or 35

53

Number of loaves of bread 5Number of bottles of milk 3Number of bottles of milk 3Number of loaves of bread 5

Example 27

Write these ratios in their simplest form:

a the number of prizes to the number of hatsb the number of hats to the number of balloons

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ExplanationSolution

a or 2 : 3

b or 3 : 49 : 12 9 3

12 4

34

6 : 9 62

93

23

The ratio is 6 : 9 or as a fraction .Simplify by dividing the numerator andthe denominator by 3.

The ratio is 9 : 12 or as a fraction .Simplify by cancelling by 3 to give thesimplest form.

912

69

9JExercise

1 Write each of the following as a ratio and as a fraction:

a the number of cats to the b the number of biros to the number of number of dogs pencils

c the number of windows to the d the number of knives to the number of number of doors forks

e the number of candles to the f the number of cats to the number of number of matches mice

g the number of coins to the h the number of frogs to the number ofnumber of notes lilypads

2 Write these ratios in their simplest form:

a 3 : 6 b 4 : 18 c 10 : 40 d 32 : 8e 45 : 30 f 12 : 36 g 4 : 4 h 18 : 2i 14 : 35 j 24 : 60 k 12 : 108 l 56 : 700

26Example

27Example

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3 Write each of the following ratios in simplest form, in the order given each time:

a 3 g to 15 g b 2 days to 16 days c 48 seconds to 36 secondsd 10 kg to 55 kg e 40C to 110C f 45 km/h to 75 km/hg 30 litres to 64 litres h 120 m to 36 m

4 Convert the amounts in each of these pairs so that they are expressed in the sameunits. Then write a ratio in its simplest form for each pair, in the order given.

a 6 days to 4 weeks b 2 kg to 50 gc 2 minutes to 45 seconds d 3 m to 70 cme 5 kilolitres to 100 litres f 4 weeks to 28 daysg $3.60 to 60 cents h 15 cm to 12 mmi $7.20 to $3.20 j 2 hours to 1000 seconds

5 In a Year 7 class of 30 students, 16 students elect to play tennis and the rest elect toplay basketball. Write each of the following ratios as a fraction and simplify it:

a the number of basketballers to the number of tennis playersb the number of tennis players to the total number of studentsc the number of basketballers to the total number of students

6 A hockey club consists of 24 senior members and 36 juniors. Write each of thefollowing ratios as a fraction and simplify it:

a the number of senior members to the total number of membersb the number of senior members to the number of juniors

7 A local tae kwon do club is selling T-shirts. There are 15 plain T-shirts, 24 T-shirts witha motif and 60 T-shirts with a picture of a person kicking. Write the following asfractions and simplify:

a the number of plain T-shirts to the number of T-shirts with motifsb the number of T-shirts with a picture to the total number of T-shirtsc the number of T-shirts with a motif to the total number of T-shirts

8 Melbourne has 4000 football coaches, Sydney has 1000 and Adelaide 2500. Writeeach of the following as a ratio and a fraction, and express it in its simplest form:

a the number of Melbourne coaches to the total number of coachesb the number of Sydney coaches to the number of Melbourne coachesc the number of Sydney coaches to the number of Adelaide coaches

Enrichment: Gear ratios

9 Here we see three gears meshed. C is the power ordrive wheel from the engine and drives the gearwheel A, which is connected to machinery. The twowheels are connected through wheel B. If the threewheels have 60, 30 and 10 cogs, draw up a table ofthe rotation of A and B for rotations of C of 12, 60,240 and 6000 turns.

Experiment with different numbers of cogs.

A

B

C

Th

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Esse

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VEL

S P

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cts

W O R K I N G

Vanishing shapesVisual effects are a feature of art and design. The use ofcolour and shape can be combined to produce imagesthat appear as though they are three-dimensional, eventhough they are drawn on a page! In this application wewill combine mathematical construction and shading to produce a three-dimensional effect.

Understanding the constructionOn a sheet of graph paper, rule a square of side 12 cm.Step 1: Divide the square into two Step 2: Divide the half-square into two equal

halves as shown. halves to make two quarters.

Step 3: Now divide the quarter-square Step 4: Continue the process until you into two equal halves. construct this shape:

Number: fractionsMathematically


1

2

116

12

12

14

of =

18=

12

14of

PL

E

Com

ICT

Th

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Creating optical illusionsIf you carefully apply shades of a chosen colour, you cancreate remarkable optical illusions. This activity can becompleted on a computer where a wide range of graduatingshades of colour are available to create this effect.

Starting with a square of side 12 cm, construct each of these fractured pictures. Include thefraction each area represents of the original square. Use colour to creat optical illusions likethat shown above.

Extending the ideaStarting with a square of side 12 cm, develop your own vanishingsquare. You could start with a different fraction or reuse one of thefractions above, but experiment with other ways of dividing thesquare.

For example, a division again based on is shown on the right.

Wall posterMake a poster of your coloured construction. Show the fraction of the original square foreach area.

12


13

13

23

of

34

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Review

Chapter summary

is called a common or proper fraction as it less than 1.

is an example of a mixed number.

is an improper fraction as it is greater than 1.

, are equivalent fractions.

Dividing the numerator and the denominator of a fraction by the same number is calledcancelling by a common factor.To compare fractions we need to convert each of them into equivalent fractions that havethe same denominator.Fractions with the same denominator are said to have common denominators.An LCM is the lowest common denominator.To add or subtract fractions they must have a common denominator.

When multiplying fractions, use the rule a d

Note:

A whole number can be written as a fraction with a denominator of 1.

Cancel before multiplying to make the calculation easier.

Simplify the answer where possible.

If you want to divide by a fraction, multiply by the reciprocal of the fraction.

The reciprocal of the fraction is the fraction .

The symbol % means per cent, which means out of 100. So 7% means 7 out of 100.We can write this as or 0.07.

Calculations with percentages are of two types:

1 We can find a percentage of a quantity.

2 We can express one quantity as a percentage of another and we can find a percentage

of a given amount.

In both cases we use the decimal or fraction equivalent to make the calculation.

7100

ba

a

b

c

b

c

d

a

b

36

24

12

,

54

214

34

5% 10% 20% 25% 33.3% 50% 66.6% 75% 100% 130%

0.05 0.1 0.2 0.25 0.33. . . 0.5 0.66. . . 0.75 1 1.3

1310

11

34

23

12

13

14

15

110

120

numerator

denominator

number of parts out of the whole

line (vinculum)

number of equal parts all together

38

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Rev

iew

Multiple-choice questions

1 Which of the following is equivalent to ?

A B C D E

2 What is the sum of and ?

A B C D E

3 What is the LCD of and ?

A 48 B 24 C 18 D 12 E 72

4 If there are three cups and four saucers in a cupboard, the ratio of cups : saucers is:

A 3 : 7 B 4 : 7 C 3 : 4 D 4 : 3 E None of these

5 of is:

A B C D E

6 Which is the odd one out?

A 40% B C 0.4 D E

7 The percentage of red balls is:

A 30% B 60% C D 3 E

8 For 2 1 , which of the following is not true?

A B C D 1 E 3.3

9 Which of the following is an improper fraction?

A B 0.5 C 30% D E

10 72% is equivalent to:

A 7.2 B C 0.072 D E1625

1823

3650

114

54

710

825

6620

19860

3310

715

14

35

610

410

400100

25

114

64

54

38

54

12

34

58

56

34

,

158

104

128

108

1012

78

34

1715

1210

125

127

145

75

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Review

1 Write these fractions in ascending order:

a b

2 Perform the calculation to write each of the following as a single fraction:

a b c d

3 Simplify:

a b c d e

f g h i j

4 Of 240 ice-creams sold by a fast-food outlet at the cricket, 80 were vanilla, 40strawberry, 32 chocolate, 16 banana and the rest mango. What percentage of the saleswas each of the flavours?

5 Of the $2750 weekly earnings from a lawn-mowing business, 32% was paid in tax,33.3% in wages, 25% in food and the rest banked. How much was banked?

6 Write these ratios as fractions in simplest form:

a 12 : 64 b 30 : 45 c 13 : 17 d 40 : 60 : 80

1 A total of 35 people attended a shopping spree.Of the 35 people, 25 decided to stop shoppingand eat lunch while the rest continued to visitthe stores.

a How many people continued shopping?

b What percentage of the group decided to

each lunch?

c What percentage shopped at some time

during the day?

d Write the ratio of those stopping for lunch compared to the whole group in its

simplest terms.

2 If hectares is to be subdivided into equal hectares lots and the remainder used

for parkland:

a How many lots can be made?

b How much land will remain for a park?

c What percentage of the overall development is parkland?

113

3434

212

178

516

1014

8 25

23

7249

13

11

112

37

59

38

25

37

318

129

125

215

118

79

914

8

211112

7

1225

15

123

, 135

, 310

, 213

23

, 56

, 58

, 113

Extended-response questions

Short-answer questions

MC

TEST

D&D

TEST

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ContentsCHAPTER 9 FractionsDo now9.1 Naming fractions9.2 Equivalent fractions9.3 Comparing fractions9.4 Adding fractions9.5 Subtracting fractions9.6 Multiplying fractions9.7 Dividing fractions9.8 Percentages9.9 Operations with percentages9.10 Ratios and fractionsWorking mathematicallyReview

Answers

9 Fractions

Documents

equivalent fractions9

improper fraction

naming fractions9

fraction andcalculate

pm page

dividing fractions9

multiplying fractions9

comparing fractions9