© boardworks ltd 2004 1 of 64 ks3 mathematics n7 percentages

© Boardworks Ltd 2004 1 of 64

KS3 Mathematics

N7 Percentages


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N7.1 Equivalent fractions, decimals and percentages

Contents

N7 Percentages

N7.2 Calculating percentages mentally

N7.3 Calculating percentages on paper

N7.4 Calculating percentages with a calculator

N7.5 Comparing proportions

N7.6 Percentage change


Many words begin with ‘cent’:

Percentages

1900 - 20001900 - 2000


“Percent” means . . .

Percentages

“out of a hundred”


A percentage is just a special type of fraction.

1% means 1 part per hundred

or100

1= 0.01

Percentages



10% means 10 parts per hundred

or10010

=101

= 0.1

Percentages




or10025

=41

= 0.25

Percentages




or10050

=21

= 0.5

Percentages




or100100

= 1

Percentages


Percentages of shapes


Estimating percentages


Equivalent fractions, decimals and percentages


Writing percentages as fractions

‘Per cent’ means ‘out of 100’. ‘Per cent’ means ‘out of 100’.

To write a percentage as a fraction we write it over a hundred.

For example,

46% =46

100Cancelling:

46100

=23

50

2350

180% =180100

Cancelling:180100

=

9

5

95

= 1 45

7.5% =7.5100

Cancelling:15

200

3

40

=340

=15

200


Writing percentages as decimals

We can write percentages as decimals by dividing by 100.

For example,

46% =46

100= 46 ÷ 100 = 0.46

7% =7

100= 7 ÷ 100 = 0.07

130% =130100

= 130 ÷ 100 = 1.3

0.2% =0.2

100= 0.2 ÷ 100 = 0.002


Percentages as fractions and decimals


Writing fractions as percentages

To write a fraction as a percentage, we can find an equivalent fraction with a denominator of 100.

85

For example,

=1720 100

× 5

× 5

and =10085

85%

1 725

= =3225

× 4

100

× 4

128and =

100128

128%


To write a fraction as a percentage you can also multiply it by 100%.

For example,38

=38

× 100%

=3 × 100%

8

25

2

=75%

2

= 3712%

Writing fractions as percentages


Writing decimals as percentages

To write a decimal as a percentage you can multiply it by 100%.

For example,

0.08 = 0.08 × 100%

= 8%

1.375 = 1.375 × 100%

= 137.5%


Using a calculator

We can also convert fractions to decimals and percentages using a calculator.

For example,

516

= 5 ÷ 16 × 100% = 31.25%

47

= 4 ÷ 7 × 100% = 57.14% (to 2 d.p.)

13 ÷ 8 × 100% = 162.5%58

=1 =138


Table of equivalences


Ordering on a number line


Dominoes


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Contents


N7 Percentages






Calculating percentages mentally

Some percentages are easy to work out mentally:

To find 1%1% Divide by 100Divide by 100





We can use percentages that we know to find other percentages.

Suggest ways to work out:

20%30%

60%

15%

2%75%

150%

49%

11%0.5 % 17.5%

Calculating percentages mentally


Spider diagram


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N7 Percentages







Calculating percentages using fractions

Remember, a percentage is a fraction out of 100.

16% of 90, means “16 hundredths of 90”.

or

16100

× 90 =16 × 90

100

4

25

18

5

= 725

= 14 25


What is 23% of 57?

We can use fractions:

23% of 57 =23

100× 57

=23 × 57

100

Working

× 20 3

50

7

1000 150

140 21

1150

+ 161

11

1

31= 1311

100

= 13 11100



What is 87% of 28?

Using fractions again:

87% of 28 =87

100× 28

=87 × 28

100

7

25

Working

87× 7

94

60

= 60925

= 24 925



Calculating percentages using decimals

We can also calculate percentages using an equivalent decimal operator.

4% of 9 = 0.04 × 9

= 4 × 9 ÷ 100

= 36 ÷ 100

= 0.36

What is 4% of 9?


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N7 Percentages








We can find more difficult percentages using a calculator.

It is always sensible when using a calculator to start by making an estimate.

For example, estimate the value of:

19% of £82 20% of £80 = £16

27% of 38m 25% of 40m =10m

73% of 159g 75% of 160g = 120g


Using a calculator

By writing a percentage as a decimal, we can work out a percentage using a calculator.

Suppose we want to work out 38% of £65.

38% = 0.38

So we key in:

0 . 3 8 × 6 5 =

And get an answer of 24.7.

We write the answer as £24.70.


We can also work out a percentage using a calculator by converting the percentage to a fraction.

Suppose we want to work out 57% of £80.

57% = 57100

= 57 ÷ 100

So we key in:


We write the answer as £45.60.

5 7 ÷ 1 0 0 × 8 0 =

Using a calculator


We can also work out percentage on a calculator by finding 1% first and multiplying by the required percentage.

Suppose we want to work out 37.5% of £59.

1% of £59 is £0.59 so, 37.5% of £59 is £0.59 × 37.5.

We key in:


We write the answer as £22.13 (to the nearest penny).

0 . 5 9 × 3 7 . 5 =

Using a calculator


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N7 Percentages







One number as a percentage of another

There are 35 sweets in a bag. Four of the sweets are orange flavour.

Start by writing the proportion of orange sweets as a fraction.

4 out of 35 =435

Then convert the fraction to a percentage.

× 100% =435

4 × 100%35

20

7=

80%7

= 1137%

What percentage of sweets are orange flavour?


Petra put £32 into a bank account. After one year she received 80p interest.

To write 80p out of £32 as a fraction we must use the same units.

In pence, Petra gained 80p out of 3200p.

803200

=1

40

We then convert the fraction to a percentage.

140

× 100% = 100%

40

5

2

= 2.5%

One number as a percentage of another

What percentage interest rate did she receive?


Using percentages to compare proportions

To compare the marks we can write each fraction as a percentage.

Matthew sat tests in English, Maths and Science.

His results were:

ScienceMathsEnglish

7480

1720

6670

Which test did he do best in?


English

7480

=7480

× 100% = 74 ÷ 80 × 100% = 92.5%

Maths

1720

=1720

× 100% = 17 ÷ 20 × 100% = 85%

Science

6670

=6670

× 100% = 66 ÷ 70 × 100% = 94.3% (to 1 d.p.)

We can see that Matthew did best in his Science test.



Nutrition Information

ChocolateCookies

Typical Value Per 10g biscuit

EnergyProteinCarbohydrateFatFibreSodium

233kj0.6g6.7g2.2g0.2g

<0.05g

Nutrition Information

CheesyCrisps

Typical Value Per 23g bag

EnergyProteinCarbohydrateFatFibreSodium

504kj1.6g13g7g

0.3g0.2g

Which product contains the smallest percentage of carbohydrate?



The chocolate cookies contain 6.7g of carbohydrate for every 10g of biscuits.

6.7g out of 10g =6.710

× 100% = 6.7 ÷ 10 × 100% = 67%

The cheesy crisps contain 13g of carbohydrate for every 23g of crisps.

13g out of 23g =1323

× 100% = 13 ÷ 23 × 100%

= 56.5% (to 1 d.p)

The cheesy crisps contain a smaller percentage of carbohydrate.



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N7 Percentages







Percentage increase and decrease

Factory workers demand 15% pay increase

SALE20% off all

marked prices!

Bus fares set to rise by 30%

PC now only

£568 Plus 17 % VAT1

2

House prices predicted to fall by 2%

next month25% extra free!


Percentage increase

There are two methods to increase an amount by a given percentage.

The value of Frank’s house has gone up by 20% since last year. If the house was worth £150 000

last year how much is it worth now?

Method 1

We can work out 20% of £150 000 and then add this to the original amount.

= 0.2 × £150 000= £30 000

The amount of the increase = 20% of £150 000

The new value = £150 000 + £30 000= £180 000


Percentage increase

We can represent the original amount as 100% like this:

100%

When we add on 20%,

20%

we have 120% of the original amount.

Finding 120% of the original amount is equivalent to finding 20% and adding it on.

Method 2

If we don’t need to know the actual value of the increase we can find the result in a single calculation.


Percentage increase

So, to increase £150 000 by 20% we need to find 120% of £150 000.

120% of £150 000 = 1.2 × £150 000

= £180 000

In general, if you start with a given amount (100%) and you increase it by x%, then you will end up with (100 + x)% of the original amount.

In general, if you start with a given amount (100%) and you increase it by x%, then you will end up with (100 + x)% of the original amount.

To convert (100 + x)% to a decimal multiplier we have to divide (100 + x) by 100. This is usually done mentally.


What happens if we increase an amount by 100%?

We take the original amount

100%

and we add on 100%.

100%

We now have 200% of the original amount.

This is equivalent to 2 times the original amount.

Percentage increase


What happens if we increase an amount by 200%?

We take the original amount

100%

and we add on 200%.

200%

We now have 300% of the original amount.

This is equivalent to 3 times the original amount.

Percentage increase


Here are some more examples using this method:

Increase £50 by 60%.

160% × £50 = 1.6 × £50

= £80

Increase £24 by 35%

135% × £24 = 1.35 × £24

= £32.40

Percentage increase

Increase £86 by 17.5%.

117.5% × £86 = 1.175 × £86

= £101.05

Increase £300 by 2.5%.

102.5% × £300 =1.025 × £300

= £307.50


Percentage decrease

There are two methods to decrease an amount by a given percentage.

A CD walkman originally costing £75 is reduced by 30% in a sale. What is the sale price?

Method 1We can work out 30% of £75 and then subtract this from the original amount.

= 0.3 × £75= £22.50

30% of £75 The amount taken off =

The sale price = £75 – £22.50= £52.50


Percentage decrease

100%

When we subtract 30%

30%

we have 70% of the original amount.

70%

Finding 70% of the original amount is equivalent to finding 30% and subtracting it.

We can represent the original amount as 100% like this:

Method 2

We can use this method to find the result of a percentage decrease in a single calculation.


Percentage decrease

So, to decrease £75 by 30% we need to find 70% of £75.

70% of £75 = 0.7 × £75

= £52.50

In general, if you start with a given amount (100%) and you decrease it by x%, then you will end up with (100 – x)% of the original amount.

In general, if you start with a given amount (100%) and you decrease it by x%, then you will end up with (100 – x)% of the original amount.

To convert (100 – x)% to a decimal multiplier we have to divide (100 – x) by 100. This is usually done mentally.


Here are some more examples using this method:

Percentage decrease

Decrease £320 by 3.5%.

96.5% × £320 = 0.965 × £320

= £308.80

Decrease £1570 by 95%.

5% × £1570 = 0.05 × £1570

= £78.50

Decrease £65 by 20%.

80% × £65 = 0.8 × £65

= £52

Decrease £56 by 34%

66% × £56 = 0.66 × £56

= £36.96


Sometimes, we are given an original value and a new value and we are asked to find the percentage increase or decrease.

Finding a percentage increase or decrease

We can do this using the following formulae:

Percentage increase =actual increase

original amount× 100%

Percentage decrease =actual decrease

original amount× 100%


Finding a percentage increase

The actual increase = 4.2 kg – 3.5 kg

= 0.7 kg

The percentage increase =0.73.5

× 100%

= 20%

A baby weighs 3.5 kg at birth. After 6 weeks the baby’s weight has increased to 4.2 kg.

What is the baby’s percentage increase in weight?


Finding a percentage decrease

All t-shirts were £24 now

only £18!

What is the percentage decrease?

The actual decrease = £24 – £18 = £6

The percentage decrease =624

× 100% = 25%1

4


Finding the original amount

Sometimes, we are given the result of a given percentage increase or decrease and we have to find the original amount.

I bought some jeans in a sale. They had 15% off and I only paid £25.50 for them.

What is the original price of the jeans?

We can solve this using inverse operations.

Let p be the original price of the jeans.

p × 0.85 = £25.50 so p = £25.50 ÷ 0.85 = £30


Sometimes, we are given the result of a given percentage increase or decrease and we have to find the original amount.

I bought some jeans in a sale. They had 15% off and I only paid £25.50 for them.

What is the original price of the jeans?

We can show this using a diagram:

Price before discount.

× 0.85%Price after discount.

÷ 0.85%




We can also use a unitary method to solve these type of percentage problems. For example,

Christopher’s monthly salary after a 5% pay rise is £1312.50. What was his original salary?

The new salary represents 105% of the original salary.

105% of the original salary = £1312.50

1% of the original salary = £1312.50 ÷ 105

100% of the original salary = £1312.50 ÷ 105 × 100

= £1250This method has more steps involved but may be easier to remember.

© boardworks ltd 2004 1 of 64 ks3 mathematics n7 percentages

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