11

Before you start this chapterPut your calculator away!1 Nikki has these fraction cards.

927

824

618

721

315

Nikki says, ‘All the fractions on the cards are equivalent fractions.’ Explain why Nikki is wrong.

2 Greg starts with one fraction, subtracts another fraction, and gets an answer of 4 __ 11 .

Suggest two fractions that Greg may be using.

3 a Work out 2 _ 3 3 1 _ 4 and 4 _ 5 3 1 _ 6 . Write your answers in their simplest form.

b Write your answers to part a as equivalent fractions with denominators of 30.

c Which is the smaller fraction, 2 _ 3 3 1 _ 4 or 4 _ 5 3 1 _ 6 ?

FractionsObjectives

This chapter will show you how to• compare fractions D

• add and subtract fractions D

• add and subtract mixed numbers C

• work out reciprocals C

• multiply and divide fractions and mixed numbers D C B

This chapter is about calculating with fractions.In Formula 1, the difference between first and second place is incredibly small. As Jenson Button said, ‘You’re qualifying on pole by half a tenth of a second.’

166 Fractions

Exercise 11A1 Put each of these sets of fractions in order of size, smallest first.

a 3 _ 4 , 10

__ 12 , 2 _ 3 b 2 _ 3 ,

17 __ 24 ,

3 _ 4 c 1 _ 4 ,

3 _ 8 ,

1 _ 3 d 2 _ 3 ,

4 _ 5 ,

11 __ 15 ,

5 _ 6

2 Which of these fractions is nearest to 1 _ 4 ?

A 1 _ 5 B 3 __ 10 C 7 __ 30 D 11 __ 40

You must show your working.

3 Derry says: I am thinking of a fraction. My fraction is bigger

than but smaller than . The numerator and the denominatorof my fraction are both single-digit numbers.

23

12

What fraction could Derry be thinking of?

AO2

D

AO3

D

D

Put these fractions in order of size, smallest first.

2 _ 3 , 3 _ 5 ,

11 __ 15

Example 1

2 __ 3 5 2 3 5 ______ 3 3 5 5 10 ___ 15

3 __ 5 5 3 3 3 ______ 5 3 3 5 9 ___ 15

3 __ 5 , 2 __ 3 , 11 ___ 15

D

Compare the fractions by writing them as equivalent fractions using a common denominator. The smallest number that 3, 5 and 15 all go into is 15, so use 15 as the common denominator.

Write each ordered fraction in its original form.

L11.1 Comparing fractions

Why learn this?Website designers use

fractions to make their pages look the same on

different size screens.

Skills check1 Work out

a 48 4 2 b 48 4 8 c 48 4 4 d 48 4 32 What is the lowest common multiple (LCM) of

a 3, 4 and 12 b 3, 5 and 15 c 2, 3 and 4 d 2, 3, 4 and 5?

ObjectivesD Compare fractions with

different denominators

Keywordscommon denominator

HELP Section 10.1

Comparing two or more fractionsTo compare fractions with different denominators, change them into equivalent fractions with the same denominator. The ‘same denominator’ is often called the common denominator.

16711.2 Adding and subtracting fractions

Exercise 11B1 Work out

a 1 _ 3 2 1 _ 4 b 2 _ 5 1 1 _ 3 c 3 _ 4 2 1 _ 5 d 1 _ 3 2 1 _ 7

e 3 _ 5 2 2 _ 7 f 3 _ 4 1 3 _ 5 g 5 _ 6 1 2 _ 7 h 1 _ 4 1 6 _ 7

2 In a driving reaction test, one driver reacted in 1 _ 8 of a second. Another driver took 2 _ 3 of a second to react. What is the difference in their reaction times?

3 Clive has two bags of flour. One bag contains 1 _ 8 kg and the other contains 1 _ 3 kg.Clive needs 1 _ 2 kg of flour to make one loaf of bread.Show that Clive has not got enough flour altogether in the two bags to make one loaf of bread.

11.2 Adding and subtracting fractions

Skills check1 Work out

a 1 _ 3 1 1 _ 3 b 5 _ 7 2 2 _ 7 c 2 _ 3 1 1 _ 6

2 Change each improper fraction into a mixed number.a 3 _ 2 b 9 _ 5 c 12

__ 7

LWhy learn this?

Carpenters often have to add or subtract

lengths measured with fractions.

ObjectivesD Add and subtract fractions

when both denominators have to be changed

Adding and subtracting fractionsTo add or subtract fractions with different denominators, write them as equivalent fractions with the same denominator, then add or subtract their numerators.

If you use the lowest common denominator, you will avoid having to simplify your answer at the end.

Keywordslowest common denominator

D

AO2

D

Work out

a 2 _ 3 1 3 _ 4 b 5 _ 6 – 1 _ 8

Example 2

a 2 __ 3 1 3 __ 4 5 8 ___ 12 1 9 ___ 12

5 8 1 9 ______ 12  5 17 ___ 12 5 1 5 ___ 12

b 5 __ 6 – 1 __ 8 5 20 ___ 24 – 3 ___ 24

5 20 – 3 _______ 24 5 17 ___ 24

Write both fractions with a common denominator.In this case 12 is the lowest common denominator (3 3 4 5 12).

Change into a mixed number at the end if you can.

In this case the lowest common denominator is 24, not 48 (6 3 8 5 48).

This answer cannot be cancelled or written as a mixed number.

D

168 Fractions

4 Greg adds together two proper fractions. Each fraction has a different denominator.He gets an answer of 1 7 __ 12 .Suggest two fractions that Greg may be using.

5 Pablo has two fraction cards. ?8

?5Both the fractions are positive.

Pablo adds the two fractions and gets an answer that simplifies to 3 __ 10 .Could Pablo’s answer be correct? Show working to support your decision.AO3

DAO2

D

11.3 Adding and subtracting with mixed numbers

Why learn this?Calculating the

distance between two villages may involve

adding mixed numbers.

Skills check1 Change each mixed number into an improper fraction.

a 4 1 _ 2 b 5 1 _ 3 c 7 2 _ 5

2 Change each improper fraction into a mixed number.a 7 _ 3 b 11

__ 2 c 13 __ 5

ObjectivesC Add and subtract mixed numbers

Adding and subtracting mixed numbersYou can add or subtract mixed numbers by changing them into improper fractions.Remember to write your answer in its lowest terms and as a mixed number if possible.

Keywordsmixed number, improper fraction

Work out

a 3 1 _ 3 1 1 3 _ 4 b 3 1 _ 5 2 1 2 _ 3

Example 3

a 3 1 __ 3 1 1 3 __ 4 5 10 ___ 3 1 7 __ 4

    5 40 ___ 12 1 21 ___ 12

    5 40 1 21 _________ 12

    5 61 ___ 12

    5 5 1 ___ 12

Change to improper fractions.

Change the answer back to a mixed number.

Write as equivalent fractions with a common denominator, then add.

C

16911.3 Adding and subtracting with mixed numbers

Exercise 11C1 Work out

a 3 2 _ 3 1 1 1 _ 4

b 3 7 __ 10 2 1 3 __ 40

c 2 1 _ 2 2 1 3 _ 4 1 5 _ 8

d 1 1 _ 9 1 1 1 _ 3 1 2 1 __ 18 2 3 1 _ 6

2 Sandra cycles from her home in Witney to Bampton.The distance is 6 1 _ 2 miles.On her way home she cycles 3 1 _ 5 miles and then her bike has a puncture.She pushes her bike the rest of the way home.How far does she push her bike?

3 Erin adds together two mixed numbers. The fractions have different denominators.Erin gets an answer of 3 17

__ 30 . Suggest two different sets of fractions that Erin may have added.

4 Patrick has two mixed-number cards. Both the mixed numbers are positive.

?61 ?

43

Patrick adds the two numbers and gets an answer that simplifies to 4 1 _ 4 .Could Patrick’s answer be correct? Show working to support your decision.

5 A small lorry is allowed to carry a maximum weight of 7 tonnes.The lorry is loaded with four pallets of machine parts.The pallets weigh 1 5 _ 6 tonnes, 1 7 _ 8 tonnes, 1 11

__ 12 tonnes and 1 1 _ 2 tonnes.Show that the lorry is overloaded.

FUNC

TIONALFUNC

TION

AL

b 3 1 __ 5 2 1 2 __ 3 5 16 ___ 5 2 5 __ 3

    5 48 ___ 15 2 25 ___ 15

    5 48 2 25 _________ 15

    5 23 ___ 15

5 1 8 ___ 15

Change to improper fractions.

Change the answer back to a mixed number.

Write as equivalent fractions with a common denominator, then subtract.

C

AO2

C

AO2

C

AO3

C

170 Fractions

L11.4 Multiplying mixed numbers

Why learn this?When decorating, you

may need to work out areas involving fractional

measurements.

Skills check1 Work out

a 3 _ 4 3 24 b 15 3 2 _ 5 c 1 _ 2 3 3 _ 5 2 True or false?

a 11 __ 3 5 3 1 _ 3 b 1 2 _ 5 5 6 _ 5 c 32

__ 9 5 3 5 _ 9

ObjectivesD Multiply a whole number by a mixed numberC Multiply a fraction by a mixed numberB Multiply a mixed number by a mixed number

Multiplying fractions and mixed numbersTo multiply mixed numbers, change them to improper fractions first.Then multiply the numerators together and multiply the denominators together.Before you multiply, cancel common factors if possible.If your answer is an improper fraction, change it back to a mixed number.

Work out

a 3 3 1 5 _ 6 b 1 _ 3 3 1 5 _ 6 c 2 1 _ 3 3 3 3 _ 4

Example 4

a 3 3 1 5 __ 6 5 3 __ 1 3 11 ___ 6

    5 1 3 11 ________ 1 3 2

    5 11 ___ 2

    5 5 1 __ 2

b 1 __ 3 3 1 5 __ 6 5 1 __ 3 3 11 ___ 6

    5 1 3 11 ________ 3 3 6

    5 11 ___ 18

c 2 1 __ 3 3 3 3 __ 4 5 7 __ 3 3 15 ___ 4

    5 7 3 5 ______ 1 3 4

    5   35 ___ 4

    5 8 3 __ 4

Change 1 5 _ 6 into an improper fraction,

then cancel.

Change 1 5 _ 6 into an improper fraction,

then multiply.

Change both mixed numbers to improper fractions, then cancel.

Change the answer back to a mixed number.

1

2

1

5

Multiply, then change the answer back to a mixed number.

DCB

17111.4 Multiplying mixed numbers

Exercise 11D1 Work out these multiplications.

Give your answers as mixed numbers when possible.

a 2 1 _ 3 3 3 b 2 2 _ 3 3 3 c 2 3 _ 4 3 3 d 2 4 _ 5 3 3

2 Which is the larger number, 10 3 1 2 _ 3 or 5 3 3 1 _ 3 ?

3 A metal rod has a mass of 4 1 _ 2 kg.

Seven of the rods are packed into a crate.

a What is the total mass of the rods?

The packing crate has a mass of 3 3 _ 4 kg.

b What is the total mass of the rods and crate?

4 A wooden door has a mass of 8 3 _ 5 kg. A packing crate has a mass of 12 1 _ 2 kg.Twenty of the doors are packed into a crate.What is the total mass of the doors and crate?

5 Work out these multiplications. Simplify your answers, and write them as mixed numbers when possible.

a 1 _ 2 3 2 1 _ 2 b 1 _ 3 3 4 1 _ 5 c 3 4 _ 9 3 3 _ 5 d 8 1 _ 4 3 2 __ 11

6 It takes 3 _ 4 of a minute to fill one bucket with water.How long does it take to fill 10 1 _ 2 buckets with water?

7 Hassan is going to pave the path in his garden.He has a rectangular path that measures 3 _ 4 m wide and 7 1 _ 2 m long.

a What is the area of the path? Area of a rectangle 5 length 3 width

Paving costs £18 per square metre. It can only be bought in whole numbers of square metres.

b What is the smallest amount that Hassan must spend in order to have enough paving for his path?

8 Caroline works at a garden nursery.It takes her 4 1 _ 2 minutes to transplant one tray of seedlings.One tray holds 15 seedlings.a How long does it take Caroline to transplant one seedling? Find 1 __ 15 of 4 1 _ 2 .

b How long does it take Caroline to transplant 200 seedlings?

9 Work out these multiplications. Simplify your answers and write them as mixed numbers.

a 1 1 _ 2 3 3 1 _ 2 b 2 1 _ 3 3 1 2 _ 5 c 4 2 _ 9 3 3 5 __ 12 d 8 1 _ 2 3 4 3 _ 4

10 A kitchen measures 4 1 _ 2 m by 3 3 _ 4 m.What is the floor area of the kitchen?

11 A large bag of flour weighs 1 1 _ 2 kg.Shani has 6 2 _ 3 bags of flour.She says that altogether she has 10 kg of flour.Is Shani correct? Show working to support your answer.

FUNC

TIONALFUNC

TION

AL

FUNC

TIONALFUNC

TION

AL

AO2

D

D

C

AO2

C

B

AO2

B

172 Fractions

12 John has a watering can that holds 6 4 _ 5 litres of water.It takes John 70 seconds (1 1 _ 6 minutes) to fill the watering can.On one Monday morning John fills the watering can 8 1 _ 2 times.

a How much water does John use on this Monday morning?

b How long does John spend filling the watering can on this Monday morning?

FUNC

TIONALFUNC

TION

AL

L11.5 Reciprocals

Why learn this?Using reciprocals

helps with multiplying and dividing fractions.

Skills check1 Work out

a 1 4 5 b 1 4 0.2 c 1 4 2.52 Change each improper fraction into a

mixed number.a 4 _ 3 b 12

__ 5 c 11 __ 6

ObjectivesC Find the reciprocal of a whole

number, a decimal or a fraction

Finding reciprocalsWhen two numbers can be multiplied together to give the answer 1, then each number is called the reciprocal of the other.

The reciprocal of a fraction is found by turning the fraction upside down.

The reciprocal of a number is 1 divided by that number.

Keywordsreciprocal

Find the reciprocal of

a 25 b 0.4 c 6 _ 7

Give your answers as mixed numbers when possible.

Example 5

a 1 ___ 25 b 1 ____ 0.4 5 1 3 10 __________ 0.4 3 10

    5 10 ___ 4

    5 5 __ 2

    5 2 1 __ 2

c 7 __ 6

5 1 1 __ 6

The reciprocal of a number is 1 divided by that number. Alternatively, think of 25 as 25 __

1 , and turn it upside

down.

Change 1 ___ 0.4

into an improper

fraction, then simplify.

Turn the fraction upside down, then change to a mixed number.

C

CExercise 11E1 Find the reciprocal of each of these numbers.

Give your answers as fractions, whole numbers or mixed numbers.a 4 b 10 c 20 d 100

e 1 _ 2 f 1 _ 5 g 2 _ 3 h 3 __ 10

AO2

B

17311.6 Dividing fractions

2 Multiply each of the numbers in Q1 by its reciprocal.What do you notice?

3 Kedar says, ‘The reciprocal of 5 __ 12 is 7 __ 12 because 5 __ 12 1 7 __ 12 5 5 1 7 ____ 12 5 1.’

Is Kedar correct? Explain your answer.

11.6 Dividing fractions

Skills check1 Cancel each fraction to its lowest terms.

a 4 __ 16 b 12 __ 15 c 32

__ 40

2 Work out a 2 _ 5 3 3 _ 7 b 3 _ 8 3 2 _ 9 c 1 1 _ 2 3 3 __ 10

Why learn this?A nurse uses fractions to

work out how many 1 _ 2 ml doses she can give from a

5 ml vial.

ObjectivesD Divide a whole number or a fraction by a fractionC Divide mixed numbers by whole numbers or fractions

B Divide mixed numbers by mixed numbers

Dividing by a fractionTo divide by a fraction, turn the fraction upside down and multiply.

If the division involves mixed numbers, change them to improper fractions first.

Work out a 18 4 2 _ 3 b 3 __ 20 4 9 __ 40

Example 6

9

1

a 18 4 2 __ 3 5 18 ___ 1 4 2 __ 3

    5 18 ___ 1 3 3 __ 2

    5 9 3 3 ______ 1 3 1

    5 27 ___ 1  5 27

b 3 ___ 20 4 9 ___ 40 5 3 ___ 20 3 40 ___ 9

    5 1 3 2 ______ 1 3 3

    5 2 __ 3

21

1 3

Multiply the numerators and multiply the denominators.

Multiply the numerators and multiply the denominators.

Turn 9 __ 40

upside down, change 4 to 3, then cancel the fractions.

Write 18 as 18 __ 1 .

Turn 2 _ 3 upside down, change 4 to 3,

and cancel the fractions.

D

AO2

C

AO3

C

174 Fractions

Exercise 11F1 Work out

a 15 4 1 _ 2 b 8 4 2 _ 5 c 3 4 2 _ 9

2 Work out

a 2 _ 3 4 4 _ 5 b 6 _ 7 4 12 __ 13 c 8 __ 11 4 24

__ 33

d 3 3 _ 4 4 10 e 1 2 _ 7 4 12 f 7 1 _ 5 4 6

3 Work out

a 2 1 _ 4 4 7 __ 12 b 1 2 _ 7 4 3 _ 4 c 3 2 _ 3 4 20 __ 21

4 Adam shares 3 1 _ 2 pizzas between four people. How much do they each receive?

5 Matthew pours 4 2 _ 3 litres of lemonade into 1 _ 4 litre glasses. How many full glasses does he pour?

Work out

a 4 1 _ 2 4 3

b 1 1 _ 2 4 3 4 _ 5

Example 7

a 4 1 __ 2 4 3 5 9 __ 2 4 3 __ 1

    5 9 __ 2 3 1 __ 3

    5 3 3 1 ______ 2 3 1

    5 3 __ 2

    5 1 1 __ 2

b 1 1 __ 2 4 3 4 __ 5 5 3 __ 2 4 19 ___ 5

    5 3 __ 2 3 5 ___ 19

    5 3 3 5 ________ 2 3 19

    5 15 ___ 38

Multiply, then change the improper fraction back to a mixed number.

Multiply the numerators and multiply the denominators.

Change both mixed numbers into improper fractions. Then turn 19 __

5

upside down and change 4 to 3.

3

1

C

BChange 4 1 _

2 into an improper fraction and

write 3 as 3 _ 1 .

Turn 3 _ 1 upside down, change 4 to 3,

and cancel the fractions.

DC

D

AO2

C

C

175Chapter 11 Review exercise

6 This is part of David’s homework.

6 1 __ 2

4 2 __ 5

Step 1: 6 4 2 __ 5

5 6 __ 1 4 2 __

5

5 6 __ 1 3 5 __

2

5 3 3 5 ______ 1 3 1

5 15 __ 1

5 15

Step 2: 1 __ 2

4 2 __ 5

5 1 __ 2

3 5 __ 2

5 1 3 5 ______ 2 3 2

5 5 __ 4

5 1 1 __ 4

Step 3: 15 1 1 1 __ 4

5 16 1 __ 4

1

3

Are his workings correct? Explain your answer.

7 Work out

a 1 1 _ 3 4 4 1 _ 2 b 2 3 _ 5 4 3 1 _ 3 c 5 4 _ 7 4 6 3 _ 8

d 4 1 _ 4 4 1 1 _ 5 e 6 3 _ 5 4 1 5 _ 7 f 4 1 _ 2 4 3 2 __ 15

8 a Work out

i 3 1 _ 2 4 4 2 _ 3 ii 5 1 _ 3 4 7 1 _ 9

Write your answers in their simplest form.

b Use your answers to part a to work out (3 1 _ 2 4 4 2 _ 3 ) 2 (5 1 _ 3 4 7 1 _ 9 ).

9 Ryan says, ‘6 2 _ 3 4 3 3 _ 4 is more than half of 3 1 _ 3 .’Is Ryan correct? Show workings to support your answer.

Review exercise1 Work out 5 4 3 _ 4 . [2 marks]

2 Brian spends 2 _ 3 of his wages on bills.He spends 1 _ 5 of his wages on clothes.The rest of his wages he saves.What fraction of his wages does Brian save? [3 marks]

3 Find the reciprocal of 0.2. Give your answer as a whole number. [2 marks]

AO2

B

AO3

B

AO2

B

B

C

AO2

D

176 Fractions

4 Emyr buys 1 1 _ 4 kg of carrots, 2 _ 3 kg of parsnips and 2 3 _ 5 kg of potatoes.What is the total weight of the vegetables that Emyr buys? [3 marks]

5 It takes 2 _ 3 of a minute to fill one jug with water.How long does it take to fill 8 1 _ 2 jugs with water?

a Give your answer in minutes as a mixed number in its simplest form. [3 marks]

b Give your answer in minutes and seconds. [1 mark]

6 Work out 4 1 _ 5 3 3 6 _ 7 .Simplify your answer and write it as a mixed number. [3 marks]

7 a Work out 4 2 _ 5 4 8 1 _ 4 .Write your answer in its simplest form. [3 marks]

b Use your answer to part a to work out the reciprocal of 4 2 _ 5 4 8 1 _ 4 .Write your answer as a mixed number in its simplest form. [2 marks]

8 The diagram shows the dimensions of Brendan’s kitchen.

5 m12

3 m78

3 m34

1 m78

Brendan buys flooring for the kitchen that costs £16.50 per square metre.The flooring can only be bought in whole numbers of square metres.How much does the flooring cost Brendan? [5 marks]

FUNC

TIONALFUNC

TION

AL

In this chapter you have learned how to

• compare fractions with different denominators D

• add and subtract fractions when both denominators have to be changed D

• multiply a whole number by a mixed number D

• divide a whole number or a fraction by a fraction D

• add and subtract mixed numbers C

• multiply a fraction by a mixed number C

• find the reciprocal of a whole number, a decimal or a fraction C

• divide mixed numbers by whole numbers or fractions C

• multiply a mixed number by a mixed number B

• divide mixed numbers by mixed numbers B

Chapter summary

AO2

C

B

AO2

B