32
Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text STRAND A: Computation 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 A2.6 More Complex Percentages A2.7 Percentage Increase and Decrease A2.8 Compound Interest and Depreciation A2.9 Reverse Percentage Problems © CIMT, Plymouth University

STRAND A: Computation A2 Using Fractions and Percentages · Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text A2.3 Quantities as Percentages To answer questions

Embed Size (px)

Citation preview

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

STRAND A: Computation

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

A2.6 More Complex Percentages

A2.7 Percentage Increase and Decrease

A2.8 Compound Interest and Depreciation

A2.9 Reverse Percentage Problems

© CIMT, Plymouth University

1© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

A2 Using Fractions andPercentages

A2.1 Fractions, Decimals and PercentagesA 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 simpleprocess. Percentages can also be converted very easily to decimals, which can be usefulwhen using a calculator. Fractions and decimals can also be converted back topercentages.

Worked Example 1

Convert each of the following percentages to fractions.

(a) 50% (b) 40% (c) 8%

Solution

(a) 5050

100% = (b) 40

40

100% = (c) 8

8

100% =

=12

=25

=225

Worked Example 2

Convert each of the following percentages to decimals.

(a) 60% (b) 72% (c) 6%

Solution

(a) 60% =60

100(b) 72% =

72100

(c) 6% =6

100

= 0 6. = 0 72. = 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(b) 0.65 =

65

100(c) 0.9 =

9

10

= 4% = 65% =90

100

= 90%

2© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

Worked Example 4

Convert each of the following fractions to percentages.

(a)3

10(b)

1

4(c)

1

3

SolutionTo convert fractions to percentages, multiply the fraction by 100%. This gives its valueas a percentage.

(a)3

10

3

10100= × % (b)

1

4

1

4100= × % (c)

1

3

1

3100= × %

= ×3

10

100

1% = ×

1

4

100

1% = ×

1

3

100

1%

=300

10% =

100

4% =

100

3%

= 30% = 25% = 3313

%

Information

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

Exercises

1. Convert each of the following percentages to fractions, giving your answers in theirsimplest 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%

2. 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%

3. 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

A2.1

3© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

4. Convert the following fractions to percentages.

(a)1

2(b)

7

10(c)

1

5(d)

3

4

(e)1

10(f)

9

10(g)

4

5(h)

4

50

(i)8

25(j)

7

20(k)

7

25(l)

2

3

5. (a) Complete the equation 2

3 15

16= =

?

?

(b) Change 7

40 to a percentage.

6. (a) Water is poured into this jug.

Copy the diagram and show accurately the waterlevel when the jug is three-quarters full.

(b) What percentage of the jug is filled with water?

7. Plan of a garden

(a) In the garden the vegetable garden has an area of 46.2 m2 . The orange field

has an area of 133.6 m2 .

What is the total area of the vegetable garden and the orange field? Giveyour answer to the nearest square metre.

(b) The garden has an area of 400 m2 .

(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 m2 . What percentage of thegarden is taken up by the pool?

A2.2 Fractions and Percentages of QuantitiesPercentages 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.

A2.1

Not to scaleOrangefield

Vegetablegarden

Lawn Pool

4© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

Worked Example 1

Find 20% of £84.

SolutionThis can be done by converting 20% to either a fraction or a decimal.

Converting to a fraction

Note that 20% = =20

10015

Therefore 20% of £84 = ×15

84£

= £ .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 increasethe price of:

(a) a bar of soap costing 90 pence (b) a packet of rice costing £2.00?

Solution

First note that 10% =1

10.

(a) 10% of 90 p = ×1

1090 p

= 9 p

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

(b) 10% of £2 = ×1

102£

= £ .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?

A2.2

5© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

Solution

First note that 25% =1

4.

25% of 500 = ×1

4500

= 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 receives5% interest. How much interest does she receive?

Solution

First convert 5% to a fraction. 5% = =5

100

1

20

5% of £20 000 = ×1

2020 000£

= £1000

So she receives £1000 interest.

Exercises

1. 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

2. Find

(a)2

5 of 80 (b)

3

4 of 120 (c)

1

5 of 90

(d)1

4 of 360 (e)

4

5 of 150 (f)

3

10 of 500

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

(a) 2 kg of powder (b) 1.2 kg of powder?

A2.2

6© CIMT, Plymouth University

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 moneydoes 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 hadincreased 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. Thebuilding company offers him a 25% discount. How much money would he saveby accepting this offer?

8. When Maria walks to school she covers a distance of 1800 m. One day shediscovers a short cut which reduces this distance by 20%. How much shorter is thenew 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 didKina 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 increasedby 60%. How much weight had the baby gained?

13. Work out

(a)7

10 of £8 (b) 20% of £25 (c)

3

8 of 6 metres.

14. (a) Calculate 15% of £600.

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

13

, 29

, 56

, 16

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

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

17. Jane wants to buy a computer costing $1800 in the USA. The deposit is2

5 of the

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.

A2.2

7© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

A2.3 Quantities as PercentagesTo 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,

3040

3040

100= × % and 4050

4050

100= × %

= 75% = 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

10. Changing this fraction to a percentage,

6

106

10100 60= × =% %

Worked Example 3

Robyn and Rachel bought a DVD for £20. Robyn paid £11 and Rachel paid £9. Whatpercentage of the total cost did each girl pay?

SolutionRobyn paid £11 out of £20, which is

11

20

11

20100 55= × =% %

Rachel paid £9 out of £20, which is

9

20

9

20100 45= × =% %

8© CIMT, Plymouth University

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?

SolutionHe saves £15 out of £200, which is

15

200

15

200100 7 5= × =% . %

Exercises1. 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 girlsand 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 000Jamaica supporters and 11 000 Trinidad supporters. What percentage of the crowdsupported each team?

6. A school won a prize of £2000. The staff spent £1600 on a new computer and therest 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 Eastendersregularly. Express this as a percentage.

9. Jamar needs another 40 stamps to complete his collection. There is a total of500 stamps in the collection. What percentage of the collection does he havealready?

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

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

A2.3

9© CIMT, Plymouth University

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 offish do people catch?

14. Two recipes for making chocolate drinks are shown in the table below.

(a) What percent of the mixture using Recipe A is chocolate?

(b) By showing suitable calculations, determine which of the two recipes, A orB, is richer in chocolate.

(c) If the mixtures from Recipe A and Recipe B are combined, what is thepercent of chocolate in the new mixture?

(d) A vendor makes chocolate drink using Recipe A. 3 cups of milk and 2 cupsof 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 tomake an overall profit of 20%?

A2.4 Addition and Subtraction of FractionsNote

The numerator is the top part of a fraction and the denominator is the bottom part of afraction.

When adding or subtracting fractions they must have the same denominator.

Worked Example 1

47

57

+ = ?

A2.3

Cups of Milk Cups of Chocolate

Recipe A 3 2

Recipe B 2 1

Investigation

The ancient Egyptians were the first to use fractions. However, they only used fractions

with a numerator of one. Thus they wrote 3

8 as

1

4

1

8+ , etc.

What do you think the Egyptians would write for the fractions 3

5,

9

20,

2

3 and

7

12?

10© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

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

47

57

97

+ =

= 12

7

Worked Example 2

3

4

2

5+ = ?

SolutionAs these fractions have different denominators, it is necessary to find the lowest commonmultiple of the denominator, that is, the smallest number into which both denominatorswill divide exactly (we sometimes refer to this as the lowest common denominator). Inthis case it is 20, since both 4 and 5 divide into 20 exactly.

3

4

2

5

15

20

8

20+ = +

=+15 8

20

=23

20

= 13

20

Worked Example 3

2

3

7

12+ = ?

SolutionIn this example, 12 is the lowest common denominator.

23

712

812

712

+ = +

=+8 7

12

=1512

= 13

12

= 114

A2.4

1

4

11© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

Worked Example 4

5

8

1

3− = ?

SolutionHere 24 is the lowest common denominator.

5

8

1

3

15

24

8

24− = −

=−15 8

24

=7

24

Exercises1. Give the answers to the following, simplifying them as far as possible.

(a)1

5

1

5+ (b)

3

8

1

8+ (c)

5

7

1

7+

(d)57

27

− (e)8

135

13− (f)

79

49

(g)7

9

8

9+ (h)

3

5

4

5+ (i)

6

7

5

7+

(j)7

103

10− (k)

89

59

− (l)4

151

15−

2. Complete each of the following.

(a)2

5

3

7 35

15

35+ = +

?(b)

1

5

1

6 30 30+ = +

? ?

=?

35=

?

30

(c)1

2

1

4 4

1

4+ = +

?(d)

3

16

5

8

3

16 16+ = +

?

=?4

=?

16

(e)47

23 21 21

+ = +? ?

(f)35

712 60 60

+ = +? ?

=?

21=

?

60

A2.4

12© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

3. Find the answers to the following, simplifying them if possible.

(a)16

38

+ (b)57

25

+ (c)18

332

+

(d)1

10

1

3+ (e)

3

7

5

8+ (f)

1

2

2

3+

(g)17

110

+ (h)58

43

+ (i)67

23

+

(j)4

7

1

2− (k)

6

11

1

4− (l)

2

3

1

6−

(m)34

23

− (n)58

512

− (o)1112

38

4. 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 2

5 cm is drilled in the middle of a metal

sheet of width 1 cm.

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

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

He wants to keep 1

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

for extra software?

9. In a school 1

3 of the students walk to school,

1

2 travel by car and the rest by

bus. What fraction of the children travel by bus?

10. A shopper buys 114

kg of cabbage and 113

kg of onions. Find the total weight

of vegetables bought.

A2.4

1 cm

25 cm

Not to scale

13© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

A2.5 Multiplication and Division of FractionsMultiplication

Consider finding 3

4 of

2

3 by starting with this rectangle.

First select 2

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

Then select 34

of the hatched area.

This area represents 3

4 of

2

3 of the original rectangle,

that is, 6

12 or

1

2 of the original rectangle.

34

of 23

is the same as 34

23

× , so

34

23

612

12

× = =

When multiplying two fractions, the numerators (top parts) should be multiplied togetherto give the numerator of the result. Similarly, the two denominators should be multipliedtogether.

In general terms, a

b

c

d

a c

b d× =

××

Worked Example 1

3

4

5

7× = ?

Solution

3

4

5

7

3 5

4 7× =

××

=1528

Worked Example 2

3

5

7

12× = ?

14© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

Solution

3

5

7

12

1 7

5 4× =

××

=7

20

Worked Example 3

11

23

4

5× = ?

Solution

11

23

4

5

3

2

19

5× = ×

=57

10

= 57

10

Worked Example 4

Calculate the exact value of

33

51

2

32

2

7×⎛

⎝⎞⎠−

Solution

33

51

2

3

18

5

5

36× = × =

so

33

51

2

32

2

76 2

2

7×⎛

⎝⎞⎠− = −

= ⎛⎝

⎞⎠

35

7

26

7or

Challenge!

The sum of 1

2,

1

3 and

1

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

The sum of 1

5,

1

6,

1

7 and

1

8 of the enrolment of School A is exactly the enrolment of

School C.

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

A2.5

15© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.5

DivisionTo understand how to divide with fractions, first consider how multiplication and divisionare 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

121

43× =

so ÷ 4 is the same as ×1

4.

Similarly, because ÷1

2 is the same as × 2 ,

61

212÷ = (check: 12

1

26× = )

and, alternatively, 6 2 12× = .

So ÷1

2 is the same as × 2 .

You can generalise these examples to give

÷ a is the same as ×1

a

÷1

bis the same as × b

and combining the two results gives

÷a

bis the same as ×

b

aFor example,

634

643

÷ = ×

= 8

(This result, that there are 8 lots of 3

4 in 6,

is shown in the diagram opposite.)

⎪⎪⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪⎪⎪

14

14

14

14

6 7444 8444

63

4× ⎫

⎬⎪⎪

⎭⎪⎪

⎬⎪⎪

⎭⎪⎪

13

13

6 7444 8444

34

67 4448 444

1

16© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.5

Similarly,

6

20

2

5

6

20

5

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

b

c

d

a

b

d

c÷ = ×

Worked Example 5

3

4

7

8÷ = ?

Solution

3

4

7

8

3

4

8

7÷ = ×

=××

3 2

1 7

=6

7

Worked Example 6

15

310

212

×

= ?

Solution

15

310

212

×

= ÷3

50

5

2

= ×3

50

2

5

=6

250

=3

125

2

1

17© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

Exercises

1. Find each of the following, cancelling when possible.

(a)3

4

5

7× (b)

1

5

7

8× (c)

4

5

1

12×

(d)3

7

9

10× (e)

4

7

5

8× (f)

6

7

3

(g)27

38

× (h)16

47

× (i)35

109

×

(j) 11

21

1

3× (k) 4

1

62

1

2× (l) 1

3

42

1

(m) 33

74

1

5× (n) 5

1

21

3

4× (o) 8

1

23

4

(p) 234

417

× (q) 538

156

× (r) 127

138

×

2. Find

(a)3

4

1

2÷ (b)

6

7

3

4÷ (c)

1

5

1

(d)3

8

4

5÷ (e)

3

7

9

10÷ (f)

7

4

2

(g) 11

4

3

4÷ (h) 5

1

2

1

4÷ (i) 1

1

72

3

(j) 412

115

÷ (k) 134

158

÷ (l) 317

178

÷

3. Find the area of each rectangle below.

(a) (b)

(c) (d)

4. 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 growingonions?

38

11

3

11

3

34

2 110

1 12

3 14 2 2

5

A2.5

18© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.5

1 12

cm1 1

2cm

1 12

cm

5. Using a calculator, or otherwise, determine the exact value of

313

235

215

6. A cube is made with sides of length 112

cm.

Find the volume and surface area of the cube.

7. A water can holds 51

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

3

4 full?

8. Calculate

13

42

1

21

1

3÷ −⎛⎝

⎞⎠

9. Find the length of the unmarked side of this rectangle if its area is 11

2 m2 .

10. A recipe requires 1

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

13

4 kg of sugar?

11. Amy runs 3 miles in 2

3 hour. What is her speed?

12. It takes a factory 3

4 hour to assemble a finished product. How many items could

be assembled in an 8 hour day?

35

m

19© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

A2.6 More Complex PercentagesNot all percentages can be expressed as simple fractions and often figures such as 4.26%may need to be used. In these cases it is often better to work with decimals.

Worked Example 1

The cost of a car is £20 000. VAT at 16.5% has to be added to this bill. Find the VATand the total bill.

SolutionUse 16.5% = 0.165.

Then16.5% of £20 000 = ×0 165 20 000. £

= £3300

So the total bill is £ £ £20 000 3300 23300+ =

Worked Example 2

Jamie has £486.27 in his building society account which earns interest of 8.21% per year.How much interest does he get and how much money does he have in his account afterthe first year?

SolutionWriting 8.21% as a decimal gives 0.0821.

8.21% of £486.27 = ×0 0821 486 27. £ .

= £ .39 92 (to the nearest p)

So at then end of the first year Jamie has

£ . £ . £ .486 27 39 92 526 19+ =in his account.

Worked Example 3

The cost of building materials for a garage and storeroom is £28 800 plus VAT at 16.5%.Find the total cost of the building materials.

Solution100% represents the original cost (£28 800).

16.5% is the increase due to VAT.

100 16 5 116 5% . % . %+ =

So the total cost can be found in one stage by finding 116.5% of £28 800.

Note that 116.5% is 1.165 as a decimal.

20© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.6

So 116.5% of £28 800 = ×1 165 28800. % £

= £33 552

The total cost of the load of blocks is £33 552.

Worked Example 4

Jessica's salary of £48 000 is to be increased by 2.5%. Find her new salary.

SolutionHer new salary is 102.5% of her old salary.

102.5% of £48 000 = ×1 025 48000. £

= £49200Her new salary is £49 200.

Worked Example 5

A holiday cruise to New York for a family costs £9995, but a special offer gives an 8.5%discount. Find the price of the cruise with the discount.

SolutionWith an 8.5% discount, 91.5% of the original price must be paid.

100 8 5 91 5% . % . %− =( )

So 91.5% of £9995 = ×0 915 9995. £

= £ .9145 43 (to the nearest p)

The discounted price is £9145.43.

Worked Example 6

Janet's gross salary is £2400 per month. Her tax-free allowances are shown below.

National Insurance 5% of gross salary

Personal Allowance £3000 per year

Calculate

(a) her gross yearly salary

(b) her total tax-free allowances for the year

(c) her taxable yearly income.

(d) A 10% tax is charged on the first £20 000 of taxable income. A 20% tax is chargedon the portion of taxable income above £20 000.Calculate the amount of income tax Janet pays.

21© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

Solution

(a) Janet's gross yearly salary = ×£2400 12

= £28 800

(b) National Insurance = ×5

10028 800£ per year

= £1440 per year

Personal Allowance = £3000 per year

Total tax-free allowances for the year = +£ £1440 3000

= £4440

(c) Taxable yearly income = −£ £28 800 4440

= £24 360

(d) Tax on first $20 000 of taxable income = ×10

10020 000£

= £2000

Tax on remaining $4360 of taxable income = ×20

1004360£

= £872

Total tax paid = +£ £2000 872

= £2872 per year

Exercises1. Find each of the following, giving your answers to the nearest pence.

(a) 32% of £50 (b) 15% of £83 (c) 12.6% of £40

(d) 4.7% of £30 (e) 6.9% of £52 (f) 3.7% of £18.62

2. (a) Add 16.5% VAT to £415.

(b) Add 3.2% interest to £1148.

(c) Increase a salary of £15 000 by 1.6%.

(d) Increase a price of £199 by 3.2%.

(e) Decrease £420 by 7%.

3. A DVD has a normal price of £15.

(a) In a sale its normal price is reduced by 12%. Find the sale price.

(b) After the sale, normal prices are increased by 2.5%. Find the new price ofthe DVD.

A2.6

22© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

4. Bacteria killed 70% of the fish in a pond. If 150 fish survived, calculate howmany fish were originally in the pond.

5. Peter earns $9000 per year in his part-time job in Boston, USA. He does not paytax on the first $3500 he earns and pays 25% tax on the rest. How much tax doeshe have to pay?

6. A man with a wife and two children,earns £32 000 a year.His annual tax-free allowancesare shown in Table 1.

Calculate

(a) his TOTAL annual tax-freeallowances

(b) his annual taxable income.

Table 2 shows the taxes that are due annually.

(c) Calculate the tax that he should pay annually.

7. A sound system costs £186 000 plus VAT at 16 12

%. Its price is increased by 4%.

How much would you have to pay to buy the sound system at the new price?

8. A French company pays a Christmas bonus of 120 euros to each of its employees.This is taxed at 25%. One year the bonus is increased by 5%. How much does anemployee take home?

9. An electricity supplier offers a 20% discount on the normal price and a further 5%discount off the normal price if customers pay directly from their banks. For onehousehold the electricity bill is normally £200. Find out how much they have topay after both discounts.

10. A mobile phone costs £35 plus VAT.

VAT is charged at 16 12

%.

How much is the VAT?

11. The usual price of a car radio is £298 plus VAT at 16 12

%.

(a) (i) Work out the exact value of 16 12

% of £298.

(ii) What is the usual price of this car radio?

A2.6

Allowance

Adult £900 each

Child £400 each

Housing £2500 per family

Table 1

Taxable Income Taxes Due

First £20 000 £1200

Remainder 30% of the remainder

Table 2

23© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.6

SUPERDVD PLAYER

£276

CASH

A discount of 15%

off the marked priceif you pay cash

BERRIES' STORESALE!1

6 OFFUSUAL PRICES

GANNET STORE

BARGAIN OFFER!

You pay NO VAT!

TERMS

A deposit of14 of the marked price

then 24 monthly paymentsof £9.45 each

12.

(a) Mr. Smith buys the DVD player for cash. How much discount is he allowed?

(b) Mr. Jones buys the DVD player on terms.

(i) How much must he pay as a deposit?

(ii) Multiply 945 by 24 without using a calculator.

Show all your working.

(iii) Work out the total price that Mr. Jones pays for his DVD player.

13.

Gannet Store and Berries' Store are selling car radios with CD players atreduced prices. The usual price of these in both stores is £466 (£400 plus£66 VAT).

(b) (i) Calculate the difference between the reduced prices in the two stores.Give your answers to the nearest cent.

Show your working clearly.

(ii) Which of the stores gives the bigger reduction?

A2.7 Percentage Increase and DecreasePercentage increases are calculated using

Percentage increase = ×actual increase

initial value100%

Similarly, percentage decreases are calculated using

Percentage decrease = ×actual decrease

initial value100%

24© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

Worked Example 1

The population of a village increased from 234 to 275 during one year. Find thepercentage increase.

Solution

Actual increase = −275 234 = 41

Percentage increase = ×41

234100%

= 17 52. % (to 2 decimal places)

Worked Example 2

When a beaker of sand is dried in a hot oven its mass reduces from 450 grams to320 grams. Find the percentage reduction in its mass.

Solution

Actual reduction = −450 320grams grams

= 130 grams

Percentage reduction = ×130

450100%

= 28 9. %

Worked Example 3

John buys pens for £5 each and then sells them to other students for £6.90. Find hispercentage profit.

Solution

Actual profit = −£ . £6 90 5

= £ .1 90

Percentage profit = ×1 90

5100

.%

= 38%

Exercises

1. A baby weighed 5.6 kg and six weeks later her weight had increased to 6.8 kg.Find the percentage increase.

2. A factory produces pens at a cost of 88 euro cents and sells them for 1.10 eurosFind the percentage profit.

3. A boat which cost £1500 was sold one year later for £999.50. Find thepercentage reduction in the value of the boat.

4. An investor bought some shares at a price of £4.88 each. The price of the sharesdropped to £3.96 each. Find the percentage loss.

A2.7

25© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: Text

5. A supermarket offers a £10 discount to all customers spending £40 or more.Kate spends £42.63 and John spends £78.82. Find the percentage saving foreach of these customers.

6. After a special offer the price of a multi-pack of baked beans was increased from£1.50 to £2.10. Find the percentage increase in the price of a multi-pack.

7. The size of a school increased so that it had 750 students instead of 680 and38 teachers instead of 37. Find the percentage increases in the number of teachersand students. Comment on your answers.

8. In a science experiment the length of a spring increased by 4 cm to 20 cm.Find the percentage increase in the length of the spring.

9. The average cost of a local telephone call for one customer dropped by 80 p to£2.40. Find the percentage reduction in the average cost of a local call.

10. In a year, the value of a house in the USA increased from $460 000 to $480 000.Find the percentage increase in the value of the house and use this to estimate thevalue after another year.

11. A battery was tested and found to power a torch for 12 hours.An improved version of the battery powered the torch for an extra 30 minutes.Find the percentage increase in the life of the batteries.

12. The value of an IT system depreciates as shown in the table.

IT System Value

New £12 000

After 1 year £10 000

After 2 years £ 8 800

After 3 years £ 8 000

During which year is the percentage decrease in the value of the IT system thegreatest?

13.Quality Garden Supplies

SUMMER SALE!

Save 20% on goods totalling£30 or more

(a) Derek bought a hose marked at £35. How much did he save?

(b) Kenton needs a large plant pot. He can buy pot A which is marked £27.95or pot B which is marked £32.45.

(i) Calculate 20% of £32.45.

(ii) How much cheaper would it be for Kenton to buy pot B than to buypot A?

A2.7

26© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.7

(c) Kenton's wife suggests that he buys pot A, together with some seeds costing£2.05 which she wants, so that he gets the 20% saving.

If he buys the seeds and pot A, express his saving as a percentage of thecost of pot A.

14.

Super Ace Games SystemSuper Ace Games System Normal Price £120

Sale Price Sale Price Sale Price Sale Price Sale Price 13 off off off off off

(a) Work out the sale price of the Super Ace Games System.

Mega Ace Games SystemMega Ace Games System Normal Price £320

Sale Price £272Sale Price £272Sale Price £272Sale Price £272Sale Price £272

(b) Find the percentage reduction on the Mega Ace Games System in the sale.

15. Leon paid £7.20 for a wallet. He sold it for £6.30. What was his loss as apercentage of the price he paid?

A2.8 Compound Interest and DepreciationSimple interest is where interest is paid at the end of the first year of an investment, thatis, on the initial amount invested (the principal). This interest, though, is not re-investedand so the principal remains the same and, if interest rates do not change, the sameamount of interest is paid again at the end of the next year.

Compound interest is when the interest paid each year is not paid out but is added to theprincipal. So the value of the account increases each year and the amount of interestearned each year will also increase. We say that the interest is compounded.

Worked Example 1

A person invests £200 in a building society account which pays 4% interest at the end ofeach year.

Find the value of the investment after 3 years if the interest is compounded.

SolutionInterest of 4% will be added at the end of each year by multiplying by 1.04.

So, value of account after 1 year: £ . £200 1 04 208× =

value of account after 2 years: £ . £ .208 1 04 216 32× =

value of account after 3 years: £ . . £ .216 32 1 04 224 97× =

Note that the amount of interest added increases each year.

The final value could have been found in one calculation:

£ . £ .200 1 04 224 973× =

27© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.8

Worked Example 2

When Gemma was born, her grandmother invested £200 in a building society for her.Find the value of this investment after 18 years if the interest rate is 6% per year and theinterest is compounded.

Solution

Final value = ×£ .200 1 0618

= £ .570 87

Problems with depreciation can be tackled in a similar way.

Worked Example 3

A boat is bought for £14 000. Its value decreases by 8% each year. Find the value of theboat after:

(a) 1 year (b) 5 years (c) 10 years.

SolutionDecreasing the value by 8% leaves 92% of the original value.

(a) Value after one year = ×£ .14 000 0 92

= £12 880

(b) Value after 5 years = ×£ .14 000 0 925

= £ .9227 14

(c) Value after 10 years = ×£ .14 000 0 9210

= £ .6081 44

Worked Example 4

A public address system has a cash price of £14 000. Under a hire purchase arrangementa 20% deposit is required, plus monthly instalments of £650 for two years.

How much money is saved by paying cash?

Solution

Deposit = ×20

10014 000£

= £2800

Total instalments = ×24 650£

= £15600

Total hire purchase price = +£ £2800 15600

= £18400

28© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.8

Amount saved by paying cash = −£ £18400 14 000

= £4400

Note

You can see from these worked examples that the total amount in an account after n years,An , with interest of r % is given by

Ar

An

n

= +⎛⎝

⎞⎠

1100 0

where A0 is the initial sum invested.

Exercises

1. Jill invests 1200 euros in a bank account which earns compound interest at the rateof 6% per annum. Find the value of her investment after:

(a) 1 year (b) 2 years (c) 5 years.

2. A sum of £5000 is to be invested for 10 years. What is the final value of theinvestment if the annual compound interest rate is:

(a) 5% (b) 4.8% (c) 7.2%?

3. Which of the following investments would earn most interest?

A £300 for 5 years at 2% compound interest per annum,

B £500 for 1 year at 3% compound interest per annum,

C £200 for 3 years at 8% compound interest per annum

4. The value of a computer depreciates at a rate of 25% per annum. Sam bought acomputer in the USA for $1600. What will be the value of the computer after:

(a) 2 years (b) 6 years (c) 10 years?

5. A car costs £9000 and depreciates at a rate of 20% per annum. Find the value ofthe car after 3 years.

6. Mr Mitchell deposited £40 000 in a bank and was paid simple interest at 7% perannum for two years.

(a) Calculate the amount of interest he will have received in total by the end ofthe two-year period.

(b) Mr Williams bought a plot of land for £40 000. The value of the landappreciated by 7% each year.

Calculate the value of the land after a period of two years.

7. If the rate of inflation were to remain constant at 3%, find what the priceof a pack of batteries, currently priced at £1.58, would be in 4 years' time.

29© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.8

8. The population of a third world country is 42 million and is growing at 2.5% perannum.

(a) What size will the population be in 3 years' time?

(b) In how many years' time will the population exceed 50 million?

9. The value of a boat depreciates at 15% per annum. A man keeps a boat for 4 yearsand then sells it.

(a) If the boat initially cost 6000 euros, find:

(i) its value after 4 years,

(ii) the selling price as a percentage of the original value.

(b) Repeat (a) for a boat which cost 12 000 euros.

(c) Comment on your answers.

10. A couple borrow £1000 to furnish their new home. They have to pay interest of18% each year on the amount they owe.

(a) Find the amount of interest which would be charged at the end of the firstyear.

(b) If they repay £300 at the end of each year, how much do they owe at theend of the third year of the loan?

A2.9 Reverse Percentage ProblemsSometimes it is necessary to reverse percentage problems. For example, if the price of atelevision set includes VAT, you might need to know how much of the price is the VAT.

Worked Example 1

The price of a computer is £1398, including VAT at 16 12

%. Find the actual cost of thecomputer and the amount of VAT which has to be paid.

SolutionTo add 16.5% VAT to a price it should be multiplied by 1.165. So to remove the VAT theprice should be divided by 1.165.

Original Price =£

.1398

1 165

= £1200

VAT = −£ £1398 1200

= £198

Worked Example 2

A customer is offered a 20% discount when buying a new bed. The discounted price is£158.40. Find the full price of the bed.

30© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.9

SolutionTo find the discounted price of the bed, the full price should be multiplied by 0.8.So to find the full price, the discounted price should be divided by 0.8.

Full price =£ .

.

158 40

0 8

= £198

Worked Example 3

Sharon invests some money in a building society at 6% interest per annum. After twoyears the value of her investment is £28 090. Find the amount she invested.

Solution

To find the final value, the amount invested would be multiplied by 1 062. .

To find the amount invested, divide the final value by 1 062. .

Amount invested =£

.

28 090

1 062

= £25 000

Exercises

1. A tourist in the USA can reclaim the VAT he has paid on the following items, theprices of which include VAT.

Camera £149.60

CD player £110.45

Watch £42.77

DVD player £406.08

(a) Find the total cost of the items without VAT at 16.5%.

(b) How much VAT can the tourist reclaim?

2. The price of a television set is £225.60 including 17.5% VAT. What would be theprice with no VAT?

3. A gas bill of £43.45 includes VAT at 8%. Find the amount of VAT paid.

4. The end-of-year profits of a company increased this year by 12% to £90 944. Findthe profits made last year.

5. A special bottle of dishwashing liquid contains 715 ml of liquid. The bottle ismarked '30% extra free'. How much liquid is there in a normal bottle?

31© CIMT, Plymouth University

Mathematics SKE, Strand A UNIT A2 Using Fractions and Percentages: TextA2.9

6. In a sale the following items are offered at discount prices as listed.

Item Sale Price Discount

DVD player £288.00 10%

TV set £373.12 12%

Computer £1124.80 24%

Calculator £13.78 5%

What were the prices of these items before the sale?

7. After one year, the value of a car has fallen by 15% to £8330. What was the valueof the car at the beginning of the year?

8. A sum is invested in a building society at 4% interest per annum and after 3 yearsthe value of the investment is £562.43. How much was originally invested?

9. Jenny's pocket money is increased by 25% each year on her birthday. When she is16 years old, her pocket money is £12.86 per week. How much did she get perweek when she was:

(a) 15 years old (b) 13 years old (c) 10 years old?

10. Jai buys a car, keeps it for 4 years and then sells it for £2100. If the value of the carhas depreciated by 12% per year, how much did Jai originally pay for the car?

Information

The Chinese represented negative numbers by indicating them in red and the Hindusdenoted them by putting a circle or a dot over the numbers. The Chinese had knowledgeof negative numbers as early as 200 BC and the Hindus as early as the 7th century.

In Europe, as late as the 16th century, some scholars still regarded negative numbers asabsurd. In 1545, Cardano (1501–1570), an Italian scholar, called positive numbers 'true'and negative numbers 'fictitious' numbers.