GCSE Mathematics for OCR Higher Student Book

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Higher

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MATHEMATICS

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GCSE for OCR Student Book

Karen Morrison, Julia Smith, Pauline McLean, Rachael Horsman and Nick Asker

© Cambridge University Press 2015

10 Fractions GCSE Mathematics for OCR (Higher)

In this chapter you will learn how to …

• recognise equivalence between fractions and mixed numbers.

• carry out the four basic operations on fractions and mixed numbers.

• work out fractions of an amount.

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or more resources relating F to this chapter, visit GCSE Mathematics Online.

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Using mathematics: real-life applications

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Nurses and other medical support staff work with fractions, decimals, percentages, rates and ratios every day. They calculate medicine doses, convert between different systems of measurement and set the patients’ drips to supply the correct amount of fluid per hour.

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“I have to work out a treatment plan for patients who are going to have radiation treatment for various cancers. The patient has to receive a certain fraction of the total dose of radiation at each treatment session.”

(Oncologist)

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Before you start …

18 24 27 28 30 32 36

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Ch 2 Check that you can find common factors of sets of numbers.

1 From this set of numbers, choose numbers that have: a a common factor of 9. b common factors 2 and 3. c common factors 3, 4, and 12. d common factors 4 and 8.

2 Choose the lowest common multiple of each set of numbers. a 5 and 10 A 15 B 50 C 10 D 20 b 8 and 12 A 12 B 96 C 36 D 24 c 2, 3 and 5 A 1 B 30 C 10 D 6

Ch 1 Know the correct order for performing operations (BIDMAS).

3 Which calculation is correct in each pair? Why?

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Ch 2 Find the lowest common multiple of sets of numbers.

Student A a

3223 62455 2 b 60 4 5 1 3 3 24 2 8 5 28 c 13 2 2 3 26 2 5 3 4 5 80

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2

2

Student B

3 2 2 3 26 2 4 5 50 2 60 4 5 1 3 3 24 2 8 5 232 13 2 2 3 26 2 5 3 4 5 5 2

© Cambridge University Press 2015

10 Fractions

Assess your starting point using the Launchpad Step 1 1 Which fraction does not belong in each set?

3 1 ___ 6 5 4 ___ ​ ​​, __ ​​ ​​ , ​​ ​​ , ​​ ___ ​​, ​​ ___ ​​ a ​ 15 5 30 35 20

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4 8 ___ 12 9 52 __ b ​ ​ ​​ , ___ ​​ ​​, ​​ ​​ , ​​ ___ ​​, ___ ​​ ​​ 7 14 21 16 91 33 18 22 11 _3 ___ ​ ___ ​​, ___ ​ c ​ ​​ ​​, ​2​ ​​, ​​ ​​, 2​​ __ 10 4 4 12 24

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Go to Section 1: Equivalent fractions

✓

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Step 2

2 Each calculation contains a mistake. Find the mistake and write the correct answer.

4 9 1 __ b ​ ​ ​​ 2 ___ ​​ ​​ 5 ___ ​ ​ ​​ 5 10 10 1 d 30 4 ​ __ ​ ​​ 5 15 2

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5 2 3 __ __ ​ ​ ​ 1 __ ​​ ​​ 5 ​ ​ ​ ​ a ​ 3 4 7 6 2 4 c ​ ​ __ ​​ 3 ​​ __ ​​ 5 ​ ___ ​ ​​ 7 5 35

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Go to Section 2: Operations with fractions

Step 3

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3 Which is greater in each pair?

3 5 a ​​ __ ​​ of 40 or ​​ __ ​​ of 60 5 8 3 7 b ​​ __ ​​ of 240 or ​​ ___ ​​ of 300 4 10 1 1 1 3 ​​ ​​ or __ ​​ ​​ of __ ​​ ​​ c ​​ __ ​​ of __ 4 2 2 4

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4 If you have read 45 pages of a 240 page book, what fraction of the book remains unread?

5 What fraction of 30 minutes is 45 seconds?

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Go to Section 3: Finding fractions of a quantity

Go to Chapter review

Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers

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GCSE Mathematics for OCR (Higher)

Section 1: Equivalent fractions Equivalent fractions represent the same value.

16 ​​ 2 ​and ___ ​ ​are equivalent. ​ 1 ,​​ __ For example, __ 4 8 64 You can find equivalent fractions by multiplying the numerator and denominator by the same number.

You can also find equivalent fractions by dividing the numerator and denominator by the same number (cancelling).

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This is known as simplifying, or reducing the fraction to its simplest terms.

When you give an answer in the form of a fraction, you usually give it in its simplest form.

6 1 4 ​​ ​ 5 1 ___ ​ 12 ​ 5 ​ __ 3​ __ ​ 5 3​ _ ​​ ​ ___ ​ 5 ​ __ ​​ ​ 30 6 18 3 27 9

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Fractions: • belong to the set of real numbers (numbers that can be found on a number line). a • are written in the form ​​ __ ​​ where b is not b equal to zero.

• are rational numbers (they can also be written as terminating or recurring decimals).

• a proper fraction is one where the numerator is smaller than the denominator.

• an improper fraction is one where the numerator is larger than the denominator; these can also be written as mixed numbers.

• a mixed number is written as an integer and a fraction, e.g. 2​​ _12 ​​is a mixed number.

When you are asked to compare fractions that look different, you can write them both with a common denominator so that you can compare them by size, or you can cross multiply to find out whether they are equivalent or not.

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Tip

Key vocabulary

Worked example 1

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common denominator: a number into which all the denominators of a set of fractions divide exactly.

Are the following pairs of fractions equivalent?

5 7 45 __ a ​ ​ ​and __ ​ ​​ b 3​​ _34 ​​ and __ ​​ ​​ 6 8 12 Method 1: Using common denominators

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20 ​and __ 21 ​ __ a ​ ​ 5 ​ 5 ​ __ ​ 7 ​ 5 ​ __ 6 24 8 24

7 ​ __ ​ 5 ​ 3 ​ __ 6

8

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Write both fractions with the same denominator by finding the lowest common multiple, and then multiplying each numerator accordingly, i.e., 6 3 4 5 24 so multiply 5 by 4 as well; 8 3 3 5 24, so multiply 7 by 3 as well.

Method 2: By cross multiplying

__ 7 ​ ​ 5 ​ ? ​__

6

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When the fractions have the same denominator it is easy to see whether they are equivalent. It is also easy to compare them by size. Write the fractions back in their original form when writing your answer.

40 ? 42

5 3 8 5 40 6 3 7 5 42

Cross multiplying a fraction converts it a c ​​ ​​ from the form __ ​​ ​​ 5 __ b d to the form ad 5 bc. If the resulting equation is not equal then the two fractions are not equivalent.

[ the fractions are not equivalent.

5 7 40 is smaller than 42, so __ ​ ​ , ​ __ ​​.

6 8 This is a useful strategy for comparing the size of fractions.

Continues on next page …

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© Cambridge University Press 2015

10 Fractions

Method 1: Using common denominators

15 Here, you can write __ ​ ​ ​with a 4 45 denominator of 12, or write ​ __ ​with a 12 denominator of 4 to find a common denominator, rather than finding the LCM.

​ 15 ​​ 3​​ __34 ​​ 5 __ 4

Write the mixed number as an improper fraction.

45 ​​ 15 ​ 3 ​ __ ​ __ 4 12

15 3 12 5 120 1 60 5 180 4 3 45 5 2 3 90 5 180

180 5 180

[ the fractions are equivalent.

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Exercise 10A

1 Determine whether the following pairs of fractions are equivalent (5) or

not (±).

3 __ a ​ ​ 2 ​and __ ​ ​​ 4 5

3 8

9 15

10 25

__ e ​ ​ ​and __ ​ ​​

5 12

c ​ ​ __ ​and __ ​ ​​

6 24

4 10

__ f ​ ​ ​and __ ​ ​​

2 11

1 10

11 9

121 99

d ​ ​ __ ​and __ ​ ​

5 20

g ​ ​ __ ​and __ ​ ​​

h ​ ​ __ ​ ​and ___ ​ ​ ​

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3 5

3 4

2 3

b ​​ __ ​and __ ​ ​​

1 4

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45 ​​ 15 ​ 5 ​ __ ​ __ 4 12

Write the mixed number as an improper fraction. Remember, to convert a mixed number to an improper fraction, you multiply the integer part by the denominator and then add the numerator.

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b 3​​ __34 ​​ 5 __ ​ 15 ​ 4

Method 2: By cross multiplying

Tip You can use the LCM of the denominators to find a common denominator, but any common denominator works (not just the lowest).

2 Find the equivalent fractions of ​​ __ ​​ with: a denominator 32 c numerator 27

b numerator 48

d denominator 52

3 How could you use cross multiplication to find the missing values in

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examples like these?

3 18 __ a ​ ​ ​ 5 ​ ___ ​​ 5

2 17

b ​​ ___ ​ 5 ​ __ ​

51

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4 Reduce the following fractions to their simplest form.

3 15 __ e ​ ​ 4 ​ 12 2 14 ____ i ​​ 2 ​​ 21

4 6 2 7 f ​​ ___ ​​ 21 18 j ​ ​ __ ​ 27

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__ a ​ ​ ​​

b ​ ​ __ ​​

25 100 36 g ​​ ____ ​​ 2 24 15 __ k ​ ​ ​​ 21

c ​ ​ ___ ​​

5 10 60 h ​ ​ ___ ​ 100 2 18 ​​ l ​​ ____ 2 42 d ​ ​ __ ​

5 Write each set of fractions in ascending order.

3 1 9 5 4 4

4 7

__ a ​ ​ ​, ​ __ ​, ​ __ ​, 1​ _34 ​, ​ __ ​​

5 3 11 19 6 4 3 24

b ​ ​ __ ​, ​ __ ​, ​ __ ​ , ​ __ ,​ 2​ _23 ​​

3 1 7 8 10 13 7 7 7 14 21 7

c ​ 2​ __ ,​ ​ __ ​, ​ __ ​, ​ __ ​, ​ __ ,​ ​ __ ​ ​

6 A mediant fraction is a fraction that lies between two other fractions.

They follow the general rule: c a 1 c a ​ ​ are two fractions, then the fraction ____ ​

​ lies between them If ​ __ ​and __ b d b 1 d c a a 1 c __ ​ , ​ ​​ . ​ ​ , ​ ____ such that __ b b 1 d d Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers

Tip Mediant fractions should not be confused with the median value in a set of data. These concepts are not related.

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GCSE Mathematics for OCR (Higher)

a Use this general rule to find a fraction between: 3 9 4 ​​ and ​ 1 ​​ and ​ __ __ ​ ​​ ii ​ ​ __ ​ ​ ​ __ i ​ 4 5 5 11 b Show how you could apply the rule to find three fractions between 3 1 ​​ and ​ __ ​ ​​. ​ __ 4 3 c Test your results to show that the answers are correct. d How does this work?

Section 2: Operations with fractions

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Find out what you can and try to explain the general rule in simple terms.

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You have already learnt in earlier school years how to add, subtract, multiply and divide fractions and mixed numbers.

Read through the examples to remind you of the key rules for operations on fractions.

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Multiplying fractions

To multiply fractions multiply the numerators and then multiply the denominators.

If possible, cancel before you multiply to make the calculations easier.

Worked example 2

2 7

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3 4

​​ ​​ a ​​ __ ​​ 3 __

3 3 2 ​​ ​5 ​ _____ 4 3 7 5 ___ ​​ 6 ​​ 28

Tip

3 14

5 ___ ​​ ​​

Give the answer in its simplest form.

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Cancelling in the first step means you don’t have to simplify the fraction to get an answer.

3 3 __ 21 __ ​​ ​​ ​​​ ​​ 2​​ ​​ 3 4 7 3 1 5 3 3 __ ​​ ​​ 3 7 5 ___ ​​ ​​ 14 2

Multiply numerators by numerators and denominators by denominators.

5 7

b ​​ __ ​​ 3 3

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5 3 3 5 ​ _____ ​ 7 3 1 15 7

5 ___ ​​ ​​

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5 2​​ __ ​​ 1 7

15 ___ ​​ ​​ cannot be simplified further but it can be

7 written as a mixed number.

3 8

c ​​ __ ​​ 3 4​​ _12 ​​

__ ​​ 3 ​​ 3 __ ​​ 9 ​​

8

2

27 16

5 ___ ​​ ​​ 5 1​​ __ ​​ 11 16

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Think of a whole number as a fraction with a denominator of 1.

Rewrite the mixed number as an improper fraction.

27 ___ ​​ ​​ cannot be simplified but it can be written as a

16 mixed number.

© Cambridge University Press 2015

10 Fractions

Adding and subtracting fractions To add or subtract fractions they must have the same denominators.

Find a common denominator and then find the equivalent fractions before you add or subtract the numerators.

Worked example 3

1 2

1 4

1 4

Use 4 as a common denominator. Write __ ​​ 1 ​​ as its 2 2 ​​. equivalent ​​ __ 4

3 4

5 __ ​​ ​​

Add the numerators.

5 6

c 2​​ _34 ​​ 2 1​​ _57 ​​

Rewrite mixed numbers as improper fractions.

Find a common denominator.

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5 5 2 6 5 ​​ 5 ___ ​​ 15 ​ 1 ​ __ 6 6 5 ___ ​​ 20 ​​ 6 5 ___ ​​ 10 ​​ or 3​​ __13 ​​ 3 5 __ ​​ ​ 1 ​ __ ​​

Add the numerators.

Simplify the answer.

11 4

12 7

Rewrite mixed numbers as improper fractions.

77 28

48 28

Find a common denominator.

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5 ___ ​​ ​ 2 ​ ___ ​​

2 Think of ​​ __ ​​ as 2 lots of 4ths, 4 1 and __ ​​ ​​ as 1 lot of 4ths. If you 4 combine them, you have 3 lots of 3 4ths, or __ ​​ ​​ . You never add the 4 denominators.

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b 2​​ _12 ​​ 1 __ ​​ ​​

Tip

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2 4

5 __ ​​ ​ 1 ​ __ ​​

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​​ ​​ a ​​ __ ​​ 1 __

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5 ___ ​​ ​ 2 ​ ___ ​​

29 28

5 ___ ​​ ​​ or 1​​ __ ​​

Subtract the numerators. Simplify and write the number as mixed number.

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1 28

Dividing fractions To divide one fraction by another fraction you multiply the first fraction by the reciprocal of the second fraction.

To find the reciprocal of a fraction you invert it.

7 4 3 4 ​​ ​​ and the reciprocal of __ ​​ ​​ is __ ​​ ​​ .

So, the reciprocal of __ ​​ ​​ is __ 4 3 4 7 The reciprocal of a whole number is a unit fraction. ​​ 1 ​​ .

For example, the reciprocal of 3 is __ ​​ 1 ​​ and the reciprocal of 12 is ___ 3 12 Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers

Key vocabulary reciprocal: the reciprocal of a

1 number, x, is 1 divided by x, i.e. ​​ __ x ​​ .

Any number multiplied by its

a reciprocal is 1. For a fraction __ ​​ ​​ , b b ​​ . the reciprocal is ​​ __

a

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GCSE Mathematics for OCR (Higher)

Worked example 4

3 4

1 2

​​ ​​ a ​​ __ ​​ 4 __ Multiply by the reciprocal of __ ​​ 1 ​​ , i.e. invert the 2 ​​ and multiply. 2 fraction to ​​ __ 1

5 __ ​​ ​​ 3 __ ​​ ​​

3 2 4 1 5 __ ​​ 64 ​​ 5 __ ​​ 32 ​​ or 1​​ __12 ​​

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b 1​​ _34 ​​ 4 2​​ _13 ​​ 5 __ ​​ ​​ 4 __ ​​ ​​

Convert mixed numbers to improper fractions.

5 __ ​​ ​ 3 ​ __ ​​

7 Multiply by the reciprocal of __ ​​ ​​ . Cancel the 7s.

3

7 4 3 __ 5 ​​ ​​ 4

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7 3

3 7

6 7

c ​​ __ ​​ 4 3 5 __ ​​ ​​ 3 __ ​​ ​​

Multiply by the reciprocal of 3.

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6 1 3 7 6 __ 5 ​​ ​​ 21 2 __ 5 ​​ ​​ 7

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7 4

The rules for order of operations and negative and positive signs also apply to calculations with fractions.

Exercise 10B 1 Simplify.

3 4

2 5

1 5

3 8

2 5

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__ e ​ ​ ​ 3 ​ __ ​ 3 ​ ___ ​​

Tip

Remember that addition and subtraction are inverse operations.

4 25

9

3 5

2

7 8

2

__ i ​ ​ ​ 3 ​ ___ ​ 3 ​ ___ ​​

1 2

1 4

1 5

1 9

2 3

2

b ​ ​ __ ​ 3 ​ __ ​​

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__ a ​ ​ ​ 3 ​ __ ​​

3 4

4 ​​ __ f ​ ​ ​ 3 ​ ___ ​ 3 ​ ___ 9 20

10 11

2

5

1 12

j ​ ​ __ ​ 3 ​ __ ​ 3 ​ __ ​​

5 8

5 7

1 5

1 2

2 3

7 10

c ​ ​ __ ​ 3 ​ __ ​​

4 11

3 7

2 7

1 2

5 1 k ​ 1_​ 29 ​ 3 1​ __ ​ 3 1​ _ ​​ 22 6

l ​ ​ __ ​ 1 ​ __ ​

1 3

15 6

m ​__ ​ ​ 1 ​ __ ​​

n ​ ​ __ ​ 2 ​ __ ​​

o ​ ​ __ ​ 2 ​ __ ​​

p ​ 4​ _34 ​ 1 ​ __ ​ ​

q ​ 8​ _25 ​ 2 3​ _12 ​​

9 r ​ 7_​ 14 ​ 2 2​ __ ​​ 10

s 9 ​ ​ _37 ​ 2 2​ _45 ​​

t ​ ​ __ ​ 4 ​ __ ​

w ​3​ _15 ​ 4 2​ _12 ​​

x ​ 1_​ 78 ​ 4 2​ _34 ​​

1 8

7 9

u ​ ​ __ ​ 4 ​ __ ​​

2 11

23

__ v ​ ​ ​ 4 ​ ___ ​​

5

1 4

7 2 What should be added to ​4​ _35 ​​to get ​9__ ​ 20 ​​?

1 6

5 4 Subtract the product of ​​ __ ​​ and ​20​ _47 ​​ from the sum of ​4_​ 79 ​​ and ​5__ ​ 18 ​​ .

© Cambridge University Press 2015

2 3

h ​ ​ __ ​ 3 ​ __ ​ 3 ​ __ ​

3 What should be subtracted from ​13​ _34 ​​to get ​5_​ 13 ​​ ?

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5 8

g ​ ​ __ ​ 3 ​ __ ​ 3 ​ __ ​​

7 9

1 4

2 3

d ​ ​ __ ​ 3 ​ __ ​

1 4

10 Fractions

5 Simplify:

1 3

Tip

c ​ ​ __ ​ 3 ​(__ ​ ​ 1 6 4 ​ __ )​ ​ 1 5 3 ​ __ ​

7 1 ​ 2 (2​ _1 ​ 2 ​ __ b 2 ​ ​ __ ​)​ 5

3 7

2 2 8 8 3 3 5 1 __ 5 1 3 5 1 d 2 ​ ​ _78 ​ 1 (8​ _14 ​ 2 6​ _38 ​ )​ e __ ​ ​ 3 ​ __ ​ 1 ​ ​ 3 ​ __ ​​ f ​​(5 4 ​ __ ​ 2 ​ __ ) ​ ​ 3 ​ __ ​ 6 4 8 3 6 11 12 5 15 5 1 3 3 2 ​ 2 ​ __ 1 ​ ​ 3 ​ __ 2 ​ g ​​(__ ​ ​ 4 ​ __ ​ )​ 2 ​(__ ​ ​ 3 ​ __ ​ ​​ h ​​(2​ _23 ​ 4 4 2 ​ __ )​ ​ 3 ​ __ ​​ i ​​(7 4 ​ __ 8 4 6 5) 9 3) 3 10 17 5 27 6 2 5 24 __ j ​(__ ​ ​ 3 ​ __ ​ )​4 ​ __ ​ 1 ​ __ ​​ k ​ 1_​ 23 ​ 4 ​ __ ​ 3 ​ __ ​ 3 ​ 1 ​ 9 35 9 3 6 35 3 3 __ 3 7 1 ​ 3 20 __ l ​ 2​ _19 ​ 1 ​ 3​ __ 5 ​ 2 ​(​ __ ​ 2 ​ ​ 2 ​ ​ __ ​ ​ __ ​ ​​ [ 27 ​ 1 ​{​ 14 35 7 ) 5 } 9 ]

Remember the rules for order of operations apply to fractions as well.

Simplify brackets first, then powers, then multiplication and/ or division, then addition and/or subtraction. When there is more than one set of brackets, work from the inner ones to the outer ones.

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6 Kevin is a professional deep-sea diver.

2 7

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2 3

a ​4 1 ​ __ ​ 3 ​ __ ​​

He needs to keep track of how much time he spends underwater to make sure he has enough air left in his tank.

2 3 and the rest is paved.

1 5

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If he spends ​9​ _34 ​​ minutes swimming to a wreck, ​12​ _56 ​​minutes exploring the wreck and ​3​ _56 ​​ minutes examining corals, how much time has he spent underwater in total?

7 In a public park, ​​ __ ​​of the area is grassed, __ ​​ ​​ is taken up by flower beds

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How much of the park area is paved?

23 8 The perimeter of a quadrilateral is ​18​ __ ​​ m.

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2 ​ 15 ​​ m, find the length of If the lengths of three sides are ​6_​ 16 ​​ m, ​7_​ 23 ​​ m and ​1__ the other side.

7 30 The water has to be removed by a piece of equipment that draws out the 3 water at a rate of ​​ __ ​​ of a litre per minute.

5 How long will it take to remove all the water?

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9 A tank contains ​​ ___ ​​ of a litre of water.

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10 There are ​1​ _34 ​​ cakes left over after a party.

These are shared out equally among six people.

What fraction does each person get?

2 15

11 If I have ​5​ _23 ​​ litres of juice, how many cups containing ___ ​​ ​​ of a litre can

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I pour?

12 Nico buys six trays of chicken pieces for his restaurant.

Each tray contains ​2​ _12 ​​ kg of chicken.

3 Each chicken meal served uses __ ​​ ​​ kg of chicken.

8 How many meals can he serve?

13 A mountaineer inserts a bolt into the rock face every ​5​ _34 ​​ metres she climbs.

If she uses 32 bolts, what is the maximum height she has climbed?

Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers

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GCSE Mathematics for OCR (Higher)

14 Ted has marked out three lengths of wood.

2 ​​of the length of B, and length B is ​1_​ 1 ​​ times as long as A. Length C is ​​ __ 3 3 97 ___ What is the length of piece C if A is ​​ ​​ m long?

3 15 Triangle ABC is isosceles with a perimeter

B

of ​12​ _34 ​​ cm.

2 11 12 cm

A

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Find the length of sides AB and AC given 11 ​​ cm. that BC is ​2​ __ 12

C

2 9

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16 What is the length of the side of a square of perimeter ​9​ _37 ​​ m?

1 6

17 On a holiday weekend Salma read ​​ __ ​​of her book on Saturday, __ ​​ ​​ on

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5 ​​ ​​ on Monday.

Sunday and ___ 12 a What fraction of the book does she still have to read?

b If there are still 49 pages left for her to read, how many pages were in the book?

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Section 3: Finding fractions of a quantity Expressing one quantity as a fraction of another

It is easy to express one quantity as a fraction of another if you remember that the numerator in a fraction tells you how many parts of the whole quantity you are dealing with.

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The denominator represents the whole quantity.

3 ​​ ​​ means you are dealing with 3 of the 5 parts that make up the whole.

So, a fraction of __ 5 Worked example 5

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a What fraction is 20 minutes of 1 hour?

b Express 35 centimetres as a fraction of a metre.

a 20 minutes is part of 60 minutes. ___ ​​ 20 ​​ 5 __ ​​ 2 ​​ 5 __ ​​ 1 ​​ of an hour.

6

3

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60

b 35 cm is part of 100 cm. ____ ​​ 7 ​​ of a metre. ​​ 35 ​​ 5 ___

100

20

20 minutes is the part of the whole, so it is the numerator.

The hour is the whole, so it is the denominator.

You cannot form a fraction using one unit for the numerator and another for the denominator, so you need to convert the hour to minutes.

35 cm is the part of the whole, so it is the numerator.

The metre is the whole, so it is the denominator.

Again, you cannot use centimetres and metres in the same fraction, so you convert 1 m to 100 cm.

To write a quantity as a fraction of another quantity make sure the two quantities are in the same units and then write them as a fraction and simplify.

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© Cambridge University Press 2015

10 Fractions

Exercise 10C 1 Calculate.

3 __ __ a __ ​​ ​​ of 12 b ​​ 1 ​​ of 45 c ​​ 2 ​​ of 36 4

3

9

5 8 3 3 3 3 1 1 __ __ g __ ​​ ​​ of __ ​​ ​​ h ​​ ​​ of ___ ​​ ​​ i ​​ 4 ​​ of ___ ​​ ​​ 2 4 3 10 9 14 3 5 __ __ j __ ​​ 1 ​​ of 2​​ _12 ​​ k ​​ ​​ of 2​​ _13 ​​ l ​​ ​​ of 3​​ _12 ​​ 4 4 6 2 Calculate the following quantities.

3 3 __ __ a __ ​​ ​​ of £28 b ​​ ​​ of £210 c ​​ 2 ​​ of £30

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3 __ __ d __ ​​ ​​ of 144 e ​​ 4 ​​ of 180 f ​​ 1 ​​ of 96

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4 5 5 __ __ d __ ​​ 2 ​​ of £18 e ​​ 1 ​​ of 3 cups of sugar f ​​ 1 ​​ of 5 cups of flour 3 2 2 3 1 1 1 __ _ __ _ __ g ​​ ​​ of 1​​ 2 ​​ cups of sugar h ​​ ​​ of 2​​ 3 ​​ cups of flour i ​​ 2 ​​ of 1​​ _12 ​​ cups of sugar 4 2 3 3 2 1 1 __ __ j __ ​​ ​​ of 4 hours k ​​ ​​ of 2​​ _2 ​​ hours l ​​ ​​ of 5 hours 4 3 3 3 3 __ ___ m __ ​​ 1 ​​ of __ ​​ ​​ of an hour n ​​ 2 ​​ of 3​​ _12 ​​ minutes o ​​ ​​ of a minute 3 4 3 15

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3 Express the first quantity as a fraction of the second.

a 12p of every £1 b 35 cm of a 2 m length

c 12 mm of 30 cm d 45 minutes per 8 hour shift e 5 minutes per hour f 150 m of a kilometre g 45 seconds of 30 minutes h 575 ml of 4 litres

4 Nick earns £18 000 per year. His friend Samir earns £24 000 per year.

What fraction of Samir’s salary does Nick earn?

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5 The floor area of a room is 12 m2. Pete buys a rug that is 110 cm wide and

160 cm long.

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What fraction of the floor area will be covered by this rug?

6 A technical college has 8400 books in their library.

3 1 ​​are general reference books, __ ​​ ​​ are technology related, ___ ​​ 4 ​​ are Of these, ​​ __ 7 7 35 engineering related and the rest are computer related.

Find the total number of books in each subject category.

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7 A section of road ​1​ _12 ​​ km long is to be tarred.

3 __ ​​ ​​ in week 2 and the rest is to be completed in week 3. ​​ 1 ​​ is tarred in week 1, __

5 6 Calculate the length of road tarred each week.

1 6 3 ​​ ​​ are travelling in ​​ 1 ​​ are travelling business class, __ travelling first class, __ 4 8 economy class and the rest are using no-frills, low-cost tickets.

8 Of 60 000 people passing through a major airport in one week, __ ​​ ​​ are

Work out the number of people travelling in each class.

Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers

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GCSE Mathematics for OCR (Higher)

Exercise 10D The Ancient Egyptians believed that anything other than a unit fraction was unacceptable (a unit fraction has a numerator of 1).

So they wrote all fractions as the sum or difference of unit fractions.

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The sum or difference always started with the largest possible unit fraction and they did not allow repetition.

1 ​​ and not as __ 1 ​​. ​​ 1 ​ 1 ​ __ ​​ 1 ​ 1 ​ __ So, for example, __ ​​ 2 ​​ would be written as __ 3 2 6 3 3 1 Try to write each of the following as the sum or difference of unit

3 5

b ​​ __ ​​

2 7

c ​​ __ ​​

2 9

d ​ ​ __ ​​

3 10

e ​​ ___ ​​

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fractions: 5 a ​​ __ ​​ 8

2 5

2 Find three unit fractions that have a sum of ​​ __ ​​when added together.

1 9 of 4 and which has three additional factors.

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3 Find a unit fraction greater than ​​ __ ​​with a denominator that is a multiple

4 Can all fractions be written as the sum or difference of unit fractions?

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Justify your answer.

Checklist of learning and understanding

Equivalent fractions

Fractions that represent the same amount are called equivalent fractions.

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You can change fractions to their equivalents by multiplying the numerator and denominator by the same value or by dividing the numerator and denominator by the same value (simplifying).

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Operations on fractions

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To add or subtract fractions, find equivalent fractions with the same denominator, then add or subtract the numerators; the denominators do not get added or subtracted.

To multiply fractions, multiply numerators by numerators and denominators by denominators.

To divide fractions, multiply the fraction being divided by the reciprocal of the divisor.

Fractions of a quantity The word ‘of’ means multiply; so to find a fraction of an amount, you multiply that amount by the fraction.

A quantity can be written as a fraction of another as long as they are in the same units. Write one quantity as the numerator and the other as the denominator and simplify.

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© Cambridge University Press 2015

10 Fractions

Chapter review

or additional questions on F the topics in this chapter, visit GCSE Mathematics Online.

1 Simplify.

15 195 18 ____ a ___ ​​ ​​ b ​​ 4​​ __ ​​ ​​ c 48 90

230

2 Write each set of fractions in ascending order.

8 4 __ 5 3 23 16 a __ ​​ ​​ , __ ​​ ​​ , ​​ ​​ , __ ​​ ​​ b 2​​ _25 ​​ , ___ ​​ ​​ , 1​​ _35 ​​ , ___ ​​ ​​ 7

9

3 Evaluate.

7 3 1 7 3 1 7 3 __ __ a __ ​​ ​ 1 ​ __ ​ 2 ​ __ ​​ b ​​ ​ 3 ​ __ ​ 1 ​ __ ​​ c ​​ ​ 4 ​ __ ​ 3 3​ 5

8

5

2

8

5

2

8

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3 1 2 __ d ​3_​ 17 ​ 1 2​ _25 ​​ e ​3__ ​ 15 ​ 2 1​ _35 ​​ f ​​ 1 ​​ of ​3__ ​ 11 ​ 1 ​ __ ​​

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9 5 6 7

4 Simplify.

m

7 4 8 3 5 3 2 2 1 __ g __ ​​ ​ 3 ​ ___ ​ 4 3​ h ​28 4 ​ __ ​ 2 ​ __ ​​ i ​​ ​​ of __ ​​ ​ 2 ​ __ ​​ 7 18 4 7 4 8 9 3 13 5 3 7 a ​​(__ ​ ​ 4 ___ ​ ​ )​​ 1 (​​ __ ​ ​ 3 __ ​ ​ )​​ b 2​​ _23 ​​ 3 ​​(8 4 __ ​ 4 ​ 1 __ ​ ​ )​​

7 8 2 ​ 2 ​ __ 1 ​ ​ 3 ​ __ 2 ​​ 11 c ​4_​ 25 ​ 1 3​ _12 ​ 1 5​ _56 ​ 2 4​ __ ​ 1 2​ _79 ​​ d ​​(7 4 ​ __ 12 9 3) 3 8

4

9

5

sa

5 a Express 425 g as a fraction of 2​​ _12 ​​ kg.

b Express 12 000 g as a fraction of 40 kg.

6 Sandy has 12​​ _12 ​​ litres of water.

3 ​​ ​​ litre can she fill?

How many bottles containing __ 4

7 A surveyor has to divide a 15 km2 area of land into equal plots each

ft

​​ 1 ​​ km2. measuring __ 2 How many plots can she make?

ra

2 8 A vertical ladder ​7​ __ ​​ m long is lowered into a manhole.

10

D

15 If ___ ​​ ​​ parts of the ladder are outside the manhole when the bottom of 32 the ladder touches the ground, how deep is the manhole?

Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers

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