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© Boardworks Ltd 2005 1 of 56
Percentages
Stage 7 Chapter
© Boardworks Ltd 2005 2 of 56
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
© Boardworks Ltd 2005 4 of 56
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
© Boardworks Ltd 2005 10 of 56
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