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Percentages

Stage 7 Chapter

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Objectives

• Solve percentage problems involving increasing and decreasing by using a multiplier

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Calculating percentages using fractions

Remember, a percentage is a fraction out of 100.

15% of 90, means “15 hundredths of 90”

or

15100

× 90 =15 × 90

100

3

20

9

2

= 272

= 13 12

Find 15% of 90

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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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Complete the activityCalculating percentages

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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% in three years. If the house was worth £150 000

three years ago, 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

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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.

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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.

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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

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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

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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.

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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.

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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

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Complete the activityPercentage increase and decrease