Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

STRAND A: NUMBER A2 Using Fractions and

Percentages Text Contents Section A2.1

Fractions, Decimals and Percentages

A2.2

Fractions and Percentages of Quantities

A2.3

Quantities as Percentages

A2.4

Addition and Subtraction of Fractions

A2.5

Multiplication and Division of Fractions

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PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

A2 Using Fractions and Percentages A2.1 Fractions, Decimals and Percentages A percentage is a means of expressing a number as a fraction of 100: the term 'percentage' simply means 'per hundred'. Converting percentages to fractions is a simple process. Percentages can also be converted very easily to decimals, which can be useful when using a calculator. Fractions and decimals can also be converted back to percentages.

Worked Example 1 Convert each of the following percentages to fractions. (a)

50%

(b)

40%

(c)

8%

Solution (a)

50% =

=

50 100

(b)

40% =

1 2

40 100

=

(c)

2 5

8% =

=

8 100

2 25

Worked Example 2 Convert each of the following percentages to decimals. (a)

60%

(b)

72%

(c)

6%

Solution (a)

60% =

60 100

(b)

72% =

= 0.6

72 100

(c)

= 0.72

6% =

6 100

= 0.06

Worked Example 3 Convert each of the following decimals to percentages. (a)

0.04

(b)

0.65

(c)

0.9

Solution (a)

0.04

=

4 100

= 4%

(b)

0.65 =

65 100

= 65%

(c)

9 10 90 = 100

0.9 =

= 90% © CIMT, Plymouth University

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A2.1

UNIT A2 Using Fractions and Percentages: Text

PRIMARY Mathematics SKE, Strand A

Worked Example 4 Convert each of the following fractions to percentages. (a)

3 10

1 4

(b)

1 3

(c)

Solution To convert fractions to percentages, multiply the fraction by 100%. This gives its value as a percentage. (a)

3 3 = × 100% 10 10

(b)

1 1 = × 100% 4 4

(c)

1 1 = × 100% 3 3

=

3 100 × % 10 1

=

1 100 × % 4 1

=

1 100 × % 3 1

=

300 % 10

=

100 % 4

=

100 % 3

= 30%

1 = 33 % 3

= 25%

Information 'Per cent' probably comes from the Latin , 'per centum', which means 'for each hundred'.

Exercises 1.

2.

3.

Convert each of the following percentages to fractions, giving your answers in their simplest form. (a)

10%

(b)

80%

(c)

90%

(d)

5%

(e)

25%

(f)

75%

(g)

35%

(h)

38%

(i)

4%

(j)

12%

(k)

82%

(l)

74%

Convert each of the following percentages to decimals. (a)

32%

(b)

50%

(c)

34%

(d)

20%

(e)

15%

(f)

81%

(g)

4%

(h)

3%

(i)

7%

(j)

18%

(k)

75%

(l)

73%

Convert the following decimals to percentages. (a)

0.5

(b)

0.74

(c)

0.35

(d)

0.08

(e)

0.1

(f)

0.52

(g)

0.8

(h)

0.07

(i)

0.04

(j)

0.18

(k)

0.4

(l)

0.3

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A2.1

UNIT A2 Using Fractions and Percentages: Text

PRIMARY Mathematics SKE, Strand A

4.

Convert the following fractions to percentages. (a) (e) (i)

5.

6.

1 2 1 10 8 25

(b) (f) (j)

7 10 9 10 7 20

(c) (g) (k)

1 5 4 5 7 25

(d) (h) (l)

3 4 4 50 2 3

2 ? 16 = = 3 15 ?

(a)

Complete the equation

(b)

Change

(a)

Water is poured into this jug. Copy the diagram and show accurately the water level when the jug is three-quarters full.

(b)

What percentage of the jug is filled with water?

7 to a percentage. 40

7.

Plan of a garden Vegetable garden Orange field

(a)

Not to scale Lawn

Pool

In the garden the vegetable garden has an area of 46.2 m 2 . The orange field has an area of 133.6 m 2 . What is the total area of the vegetable garden and the orange field? Give your answer to the nearest square metre.

(b)

The garden has an area of 400 m 2 . (i)

The lawn is 30% of the garden. Calculate the area of the lawn.

(ii)

A pool in the garden has an area of 80 m 2 . What percentage of the garden is taken up by the pool?

A2.2 Fractions and Percentages of Quantities Percentages are often used to describe changes in quantities or prices. For example, '30% extra free'

'10% discount'

'add 16 12 % VAT'

This section deals with finding fractions or percentages of quantities.

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A2.2

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

Worked Example 1 Find 20% of £84.

Solution This can be done by converting 20% to either a fraction or a decimal. Converting to a fraction

20 1 = 100 5

Note that

20% =

Therefore

20% of £84 =

1 × £84 5

= £16.80

Converting to a decimal Note that

20% = 0.2

Therefore

20% of £84 = 0.2 × £84

= £16.80

Worked Example 2 A shopkeeper decides to increase some prices by 10%. By how much would she increase the price of: (a)

a bar of soap costing 90 pence

(b)

a packet of rice costing £2.00?

Solution First note that 10% = (a)

1 . 10

1 × 90 p 10 = 9p

10% of 90 p =

So the cost of a bar of soap will be increased by 9 pence (b)

10% of £2

=

1 × £2 10

= £0.20 or 20 p

So the cost of a packet of rice is increased by 20 pence.

Worked Example 3 A farmer decides to sell 25% of his herd of 500 cattle. How many cows does he sell?

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A2.2

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

Solution First note that 25% =

1 . 4

25% of 500 =

1 × 500 4

= 125

So he sells 125 cows.

Worked Example 4 Natasha invests £20 000 in a building society account. At the end of the year she receives 5% interest. How much interest does she receive?

Solution 5% =

First convert 5% to a fraction.

5 1 = 100 20

5% of £20 000 =

1 × £20 000 20

= £1000 So she receives £1000 interest.

Exercises 1.

2.

Find (a)

10% of 200

(b)

50% of £5

(c)

20% of £8

(d)

25% of £10 000

(e)

40% of £500

(f)

90% of 200

(g)

33 13 % of £12

(h)

75% of 800

(i)

75% of 1000

(j)

80% of 20 kg

(k)

70% of 5 kg

(l)

30% of 50 kg

(m)

5% of 100 m

(n)

20% of 50 m

(o)

25% of £30

Find (a) (d)

3.

2 of 80 5 1 of 360 4

3 of 120 4 4 of 150 5

(b) (e)

(c) (f)

1 of 90 5 3 of 500 10

A firm decides to give 20% extra free in their packets of soap powder. How much extra soap powder would be given away free with packets which normally contain (a)

2 kg of powder

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(b)

1.2 kg of powder?

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A2.2

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

4.

A picture costs £30 000. A buyer is given a 10% discount. How much money does the buyer save?

5.

John has invested £500 in a building society. He gets 5% interest each year. How much interest does he get in a year?

6.

Laura bought an antique vase for £12 000. Two years later its value had increased by 25%. What was the new value of the vase?

7.

Kevin wants to extend his house. The cost of the extension is £24 000. The building company offers him a 25% discount. How much money would he save by accepting this offer?

8.

When Maria walks to school she covers a distance of 1800 m. One day she discovers a short cut which reduces this distance by 20%. How much shorter is the new route?

9.

Chen earns £3000 per year from his part-time job. He is given a 5% pay rise. How much extra does he earn each year?

10.

George weighed 90 kg. He went on a diet and tried to reduce his weight by 10%. How many kilograms did he try to lose?

11.

Kina's mother decided to increase her pocket money by 40%. How much extra did Kina receive each week if previously she had been given £4.00 per week?

12.

A newborn baby girl weighed 4 kg. In the first three months her weight increased by 60%. How much weight had the baby gained?

13.

Work out

14.

(a)

7 of £8 10

(a)

Calculate 15% of £600.

(b)

List these fractions in order of size, starting with the smallest.

(b)

20% of £25

(c)

3 of 6 metres. 8

1 2 5 1 , , , 3 9 6 6 15.

In a certain school, 58% of the students are girls. If there are 406 girls in the school, calculate the total number of students in the school.

16.

An athletics stadium has 35 000 seats. 4% of the seats are fitted with headphones to help people hear the announcements. How many headphones are there in the stadium?

17.

Jane wants to buy a computer costing $1800 in the USA. The deposit is price of the computer. Jane's father gives her 30% of the price. Will this be enough for her deposit? You must explain your answer fully.

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2 of the 5

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

A2.3 Quantities as Percentages To answer questions such as, In a test, is it better to score 30 out of 40 or 40 out of 50? it is helpful to express the scores as percentages.

Worked Example 1 Express '30 out of 40' and '40 out of 50' as percentages. Which is the higher score?

Solution '30 out of 40' can be written as

30 40 and '40 out of 50' can be written as . 40 50

Changing these fractions to percentages,

30 30 = × 100% 40 40

and

= 75%

40 40 = × 100% 50 50 = 80%

So '40 out of 50' is the higher score, since 80% is greater than 75%.

Worked Example 2 A student scores 6 out of 10 in a test. Express this as a percentage.

Solution '6 out of 10' can be written as

6 . Changing this fraction to a percentage, 10 6 6 = × 100% = 60% 10 10

Worked Example 3 Robyn and Rachel bought a DVD for £20. Robyn paid £11 and Rachel paid £9. What percentage of the total cost did each girl pay?

Solution Robyn paid £11 out of £20, which is 11 11 = × 100% = 55% 20 20

Rachel paid £9 out of £20, which is 9 9 = × 100% = 45% 20 20 © CIMT, Plymouth University

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A2.3

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

Worked Example 4 David earns £200 per week and saves £15 towards the cost of a mobile phone. What percentage of his earnings does he save?

Solution He saves £15 out of £200, which is 15 15 = × 100% = 7.5% 200 200

Exercises 1.

Express each of the following as percentages. (a)

8 out of 50

(b)

3 out of 25

(c)

8 out of 20

(d)

3 out of 10

(e)

6 out of 50

(f)

6 out of 40

(g)

12 out of 80

(h)

9 out of 30

(i)

27 out of 30

(j)

120 out of 300

(k)

84 out of 200

(l)

260 out of 400

(m)

28 out of 70

(n)

18 out of 60

(o)

51 out of 60

2.

In a class of 25 students there are 10 girls. What percentage of the class are girls and what percentage are boys?

3.

In the USA, the price of a bar of chocolate is 25 cents and includes 5 cents profit. Express the profit as a percentage of the price.

4.

The value of a house is £40 000 and the value of the contents is £3200. Express the contents value as a percentage of the house value.

5.

In the crowd at a cricket match between Jamaica and Trinidad there were 14 000 Jamaica supporters and 11 000 Trinidad supporters. What percentage of the crowd supported each team?

6.

A school won a prize of £2000. The staff spent £1600 on a new computer and the rest on software. What percentage of the money was spent on software?

7.

A book contained 80 black and white pictures and 120 colour pictures. What percentage of the pictures were in colour?

8.

In a survey of 300 people it was found that 243 people watched Eastenders regularly. Express this as a percentage.

9.

Jamar needs another 40 stamps to complete his collection. There is a total of 500 stamps in the collection. What percentage of the collection does he have already?

10.

A 600 ml bottle of shampoo contains 200 ml of free shampoo. What percentage is free?

11.

Adrian finds that in a delivery of 500 bricks there are 20 broken bricks. What percentage of the bricks are broken?

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A2.3

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

12.

A glass of drink contains 50 ml of fruit juice and 200 ml of lemonade. What percentage of the drink is lemonade?

13.

Research shows that there are 20 000 different types of fish in the world. People catch only 9000 different types. What percentage of the different types of fish do people catch?

14.

Two recipes for making chocolate drinks are shown in the table below. Cups of Milk

Cups of Chocolate

Recipe A

3

2

Recipe B

2

1

(a)

What percent of the mixture using Recipe A is chocolate?

(b)

By showing suitable calculations, determine which of the two recipes, A or B, is richer in chocolate.

(c)

If the mixtures from Recipe A and Recipe B are combined, what is the percent of chocolate in the new mixture?

(d)

A vendor makes chocolate drink using Recipe A. 3 cups of milk and 2 cups of chocolate can make 6 bottles of chocolate drink. A cup of milk costs £0.70 and a cup of chocolate costs £1.15. (i)

What is the cost of making 150 bottles of chocolate drink?

(ii)

What should be the selling price of each bottle of chocolate drink to make an overall profit of 20%?

Investigation The ancient Egyptians were the first to use fractions. However, they only used fractions with a numerator of one. Thus they wrote

3

as

8

1 4

+

1

, etc.

8

What do you think the Egyptians would write for the fractions

3 5

,

9 20

,

2 3

and

7 12

?

A2.4 Addition and Subtraction of Fractions Note The numerator is the top part of a fraction and the denominator is the bottom part of a fraction. When adding or subtracting fractions they must have the same denominator.

Worked Example 1 4 5 + = ? 7 7 © CIMT, Plymouth University

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A2.4

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

Solution As both fractions have the same denominator (7), they can simply be added to give

4 5 9 + = 7 7 7 = 1

2 7

Worked Example 2 3 2 + = ? 4 5

Solution As these fractions have different denominators, it is necessary to find the lowest common multiple of the denominator, that is, the smallest number into which both denominators will divide exactly (we sometimes refer to this as the lowest common denominator). In this case it is 20, since both 4 and 5 divide into 20 exactly. 3 2 15 8 + = + 4 5 20 20 =

15 + 8 20

23 20 3 = 1 20

=

Worked Example 3 2 7 + = ? 3 12

Solution In this example, 12 is the lowest common denominator.

2 7 8 7 + = + 3 12 12 12 =

8+7 12

=

15 12

3 1 12 4 1 = 1 4

= 1

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A2.4

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

Worked Example 4 5 1 − = ? 8 3

Solution Here 24 is the lowest common denominator. 5 1 15 8 − = − 8 3 24 24 15 − 8 24 7 = 24 =

Exercises 1.

2.

Give the answers to the following, simplifying them as far as possible. (a)

1 1 + 5 5

(b)

3 1 + 8 8

(c)

5 1 + 7 7

(d)

5 2 − 7 7

(e)

8 5 − 13 13

(f)

7 4 − 9 9

(g)

7 8 + 9 9

(h)

3 4 + 5 5

(i)

6 5 + 7 7

(j)

7 3 − 10 10

(k)

8 5 − 9 9

(l)

4 1 − 15 15

Complete each of the following. (a)

2 3 ? 15 + = + 5 7 35 35 =

(c)

? 35

(d)

? 4

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(f)

? 21

? 16

3 7 ? ? + = + 5 12 60 60 =

11

? 30

3 5 3 ? + = + 16 8 16 16

=

4 2 ? ? + = + 7 3 21 21 =

1 1 ? ? + = + 5 6 30 30 =

1 1 ? 1 + = + 2 4 4 4

= (e)

(b)

? 60

A2.4

UNIT A2 Using Fractions and Percentages: Text

PRIMARY Mathematics SKE, Strand A

3.

4.

Find the answers to the following, simplifying them if possible. (a)

1 3 + 6 8

(b)

5 2 + 7 5

(c)

1 3 + 8 32

(d)

1 1 + 10 3

(e)

3 5 + 7 8

(f)

1 2 + 2 3

(g)

1 1 + 7 10

(h)

5 4 + 8 3

(i)

6 2 + 7 3

(j)

4 1 − 7 2

(k)

6 1 − 11 4

(l)

2 1 − 3 6

(m)

3 2 − 4 3

(n)

5 5 − 8 12

(o)

11 3 − 12 8

A garden has an area of

2 5

acre. The owner buys an extra

1 3

acre of land to

increase the size of the garden. What is the new size of the garden?

5.

A company makes a profit of £

3 4

million in one year and £

2 3

million the

next year. Find the total profits for the two-year period.

6.

A hole of radius

Not to scale

2 5

cm is drilled in the middle of a metal

sheet of width 1 cm. 2 5

cm

How far is it from the edge of the sheet to the hole?

1 cm

7.

A council decides to turn

1 3

of a park into a dog-free zone. It later bans dogs from

the play area which occupies

1 10

of the park and which was originally outside the

dog-free zone. What fraction of the park is now open to dogs? 8.

Mike has filled

3 5

He wants to keep

of the space on the hard disc in his computer with software. 1 4

of the disc free from software. What fraction of the disc is left

for extra software?

1 1 of the students walk to school, travel by car and the rest by 3 2 bus. What fraction of the children travel by bus?

9.

In a school

10.

A shopper buys 1

1 1 kg of cabbage and 1 kg of onions. Find the total weight 4 3 of vegetables bought.

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PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

A2.5 Multiplication and Division of Fractions Multiplication Consider finding

First select

3 2 of by starting with this rectangle. 4 3

2 of the rectangle, as shown by the hatched area. 3

3 of the hatched area. 4 3 2 This area represents of of the original rectangle, 4 3 6 1 that is, or of the original rectangle. 12 2 Then select

3 2 3 2 of is the same as × , so 4 3 4 3 3 2 6 1 × = = 4 3 12 2 When multiplying two fractions, the numerators (top parts) should be multiplied together to give the numerator of the result. Similarly, the two denominators should be multiplied together. In general terms,

a c a×c × = b d b×d

Worked Example 1 3 5 × = ? 4 7

Solution 3 5 3×5 × = 4 7 4×7 15 = 28

Worked Example 2 3 7 × = ? 5 12

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A2.5

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

Solution 3 7 1× 7 × = 5 12 5×4 7 = 20

Worked Example 3 1 4 ×3 = ? 2 5

1

Solution 1

1 4 3 19 ×3 = × 2 5 2 5 57 = 10 7 = 5 10

Worked Example 4 Calculate the exact value of

⎛3 3 × 1 2⎞ − 2 2 ⎝ 5 3⎠ 7

Solution 3

so

3 2 18 5 ×1 = × =6 5 3 5 3

⎛3 3 × 1 2⎞ − 2 2 = 6 − 2 2 ⎝ 5 3⎠ 7 7 = 3

5 ⎛ 26 ⎞ or 7 ⎝ 7⎠

Challenge! The sum of The sum of School C.

1 2 1 5

, ,

1 3 1 6

and

1

of the enrolment of School A is exactly the enrolment of School B.

4

,

1 7

and

1 8

of the enrolment of School A is exactly the enrolment of

What are the enrolments of these three schools, assuming that no school has more than 1000 students?

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A2.5

UNIT A2 Using Fractions and Percentages: Text

PRIMARY Mathematics SKE, Strand A

Division To understand how to divide with fractions, first consider how multiplication and division are related. Take as an example, 3 × 4 = 12

Then it is also true that 12 ÷ 4 = 3

We say that ' × 4 ' and ' ÷ 4 ' are inverse (reversed or opposite) operations. Note that 12 ×

so ÷ 4 is the same as ×

1 = 3 4

1 . 4

1 is the same as × 2 , 2 1 6÷ = 12 2 and, alternatively, 6 × 2 = 12 .

Similarly, because ÷

So ÷

(check: 12 ×

1 = 6) 2

1 is the same as × 2 . 2

You can generalise these examples to give 1 a

÷a

is the same as

×

1 b

is the same as

×b

÷

and combining the two results gives

÷

a b

3 4 444 64447 8

is the same as

×

b 1 a

For example,

3 4 6÷ = 6× 4 3 = 8 6×

(This result, that there are 8 lots of is shown in the diagram opposite.)

3 in 6, 4

3 4

1 4

1 4

1 4

644474448 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩

644474448

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15

1 4

⎫ ⎪ ⎪ ⎬1 × ⎪ ⎪⎭ ⎫ ⎪ ⎪ ⎬1 × ⎪ ⎪⎭

3 4

3 4

A2.5

PRIMARY Mathematics SKE, Strand A

UNIT A2 Using Fractions and Percentages: Text

Similarly, 6 2 6 5 ÷ = × 20 5 20 2 =

3 4

So to divide by a fraction, the fraction should be inverted, that is, turned upside down, and then multiplied. In general terms,

a c a d ÷ = × b d b c

Worked Example 5 3 7 ÷ = ? 4 8

Solution 3 7 3 8 ÷ = × 4 8 7 14 =

3×2 1× 7

=

6 7

2

Worked Example 6 1 3 × 5 10 = ? 1 2 2

Solution 1 3 × 5 10 1 2 2

=

=

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3

÷

50

3 50

×

=

6 250

=

3 125

5 2

2 5

A2.5

UNIT A2 Using Fractions and Percentages: Text

PRIMARY Mathematics SKE, Strand A

Exercises 1.

2.

3.

Find each of the following, cancelling when possible. (a)

3 5 × 4 7

(b)

1 7 × 5 8

(c)

4 1 × 5 12

(d)

3 9 × 7 10

(e)

4 5 × 7 8

(f)

6 3 × 7 4

(g)

2 3 × 7 8

(h)

1 4 × 6 7

(i)

3 10 × 5 9

(j)

1

1 1 ×1 2 3

(k)

4

1 1 ×2 6 2

(l)

1

3 1 ×2 4 7

(m)

3

3 1 ×4 7 5

(n)

5

1 3 ×1 2 4

(o)

8

1 4 ×3 2 7

(p)

2

3 1 ×4 4 7

(q)

5

3 5 ×1 8 6

(r)

1

2 3 ×1 7 8

Find (a)

3 1 ÷ 4 2

(b)

6 3 ÷ 7 4

(c)

1 1 ÷ 5 7

(d)

3 4 ÷ 8 5

(e)

3 9 ÷ 7 10

(f)

7 2 ÷ 4 5

(g)

1

1 3 ÷ 4 4

(h)

5

1 1 ÷ 2 4

(i)

1

1 3 ÷2 7 8

(j)

4

1 1 ÷1 2 5

(k)

1

3 5 ÷1 4 8

(l)

3

1 7 ÷1 7 8

Find the area of each rectangle below. (a)

(b) 3 4

3 8 1

1 12

13

(c)

(d)

2 101

1

13

3 14

4.

2 25

Mary has a garden. She grows vegetables on

1 2

of her garden. On

1 4

of this

vegetable area she grows onions. What fraction of the garden is used for growing onions?

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A2.5

PRIMARY Mathematics SKE, Strand A

5.

Using a calculator, or otherwise, determine the exact value of 3

6.

UNIT A2 Using Fractions and Percentages: Text

1 3 − 2 3 5 1 2 5

A cube is made with sides of length 1

1 cm. 2

1 12 cm

1 12

1 12 cm

cm

Find the volume and surface area of the cube.

7.

A water can holds 5

8.

Calculate

1

9.

1 2

litres when full. How much water is in the can if it is

Find the length of the unmarked side of this rectangle if its area is 1

A recipe requires

1

3 4

4

full?

3 ⎛ 1 1 ÷ 2 −1 ⎞ 4 ⎝ 2 3⎠

3 5

10.

3

1

1 2

m2 .

m

kg of sugar for a cake. How many cakes could be made with

4

kg of sugar?

11.

Amy runs 3 miles in

12.

It takes a factory

3 4

2 3

hour. What is her speed?

hour to assemble a finished product. How many items could

be assembled in an 8 hour day?

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