# Percentages

Here is everything you need to know about percentages for GCSE maths (Edexcel, AQA and OCR). You’ll learn how to find the percentage of an amount and calculate with percentage multipliers.

You will also work out how to increase and decrease a number by a percentage, percentage change and reverse percentages.

Look out for the percentages worksheets and exam questions at the end.

## What are percentages?

A percentage is a number which is expressed as a fraction of 100

Percent means “number of parts per hundred” and the symbol we use for percent is the percent sign %.

E.g.

$43\%=\frac{43}{100}=0.43$

$1\%=\frac{1}{100}=0.01$

### What are percentages?

There are different types of percentage questions.

• Percentage of an amount

This is where we are asked to find a certain percentage of an amount.

E.g. Find 25% of £32

This is the same as finding a ¼ of £32.

$32\div4=8$

Step-by-step guide: Percentage of an amount

• Percentage multipliers

This is where we can find a decimal number and use it as a multiplier to make calculating percentages more efficient.

E.g. Find 41% of £800

We can use 0.41 as a multiplier to find the amount needed.

$41\%=\frac{41}{100}=0.41$

$800\times0.41=328$

Step-by-step guide: Percentage multipliers

• Percentage increase

This is where we are asked to increase (make bigger) a value by a certain amount.

E.g. Increase 40g by 10%.

We can find 10% and add it on.

10% of 40 is 4

$40+4=44$

Step-by-step guide: Percentage increase

• Percentage decrease

This is where we are asked to decrease (make smaller) a value by a certain amount.

E.g. Decrease 60kg by 10%.

We can find 10% and subtract it.

10% of 60 is 6

$60-6=54$

Step-by-step guide: Percentage decrease

• Percentage change

When values change, we can express this change as a percentage of the original value.

E.g. Work out the percentage change of 26kg from 25kg.

The actual change from 25 to 26 is 1.

$\frac{1}{25}=\frac{4}{100}=4\%$

Step-by-step guide: Percentage change

• Reverse percentages

This is where we are given a certain percentage of a number and we have to find the original number.

E.g. 20% of a number is 6, what is the number?

$20\%=\frac{1}{5}$

To find the original number we need to find 100% or one whole.

$5\times6=30$

Step-by-step guide: Reverse percentages

## How to use percentages

In order to use percentages, you need to be clear about what you have and what you are trying to work out.  Percentage questions can vary and there will be links in the other sections with more details.

1. Write down what you have and what you are trying to find.
2. Work out what you need.
3. Write down the final answer.

## Percentages examples

### Example 1: percentage of an amount

Find 23% of £160.

1. Write down what you have and what you are trying to find.

100% is £160.

We need to work out 23% of £160.

2 Work out what you need.

$23\% = 20\%+3\%$

$10\% = 16, 1\% = 1.6$

$(2\times16)+(3\times1.6)=32+4.6=36.8$

3 Write down the final answer.

23% of £160 is £36.80

### Example 2: percentage multipliers

Find 39% of £4700

100% is £4700

We want to find 39% of £4700

We can write the percentage as a decimal number.

$39\%=0.39$

This gives the percentage in decimal form which is the percentage multiplier.

$4700\times0.39=1833$

39% of £4700 is £1833

### Example 3: percentage increase

Increase £200 by 30%

100% is £200

We want to find 130% of £200

$100\%+30\%=130\%$

10% is 20

$200\div10=20$

30% is 60

$3\times20=60$

So 130% is

$200 + 60 = 260$

OR

You could also use the decimal multiplier.

$100+30=130$

$130\%=\frac{130}{100}=1.30$

$200\times1.3=260$

£200 increased by 30% is £260

### Example 4: percentage decrease

Decrease 700g by 20%

100% is 700 g

We want to find 80% of 700 g

$100\%-20\%=80\%$

10% is 70

$700\div10=70$

20% is 140

$2\times70=140$

So 80% is

$700-140=560$

OR

You could use the decimal multiplier.

$100-20=80$

$80\%=\frac{80}{100}=0.80$

$700\times0.80=560$

700g decreased by 20% is 560 g.

### Example 5: percentage change

The price of a t-shirt increased from £10 to £12.  Work out the percentage change.

The original amount is £10.

We need to find the change as a percentage of the original amount.

The actual change is

$12-10=2$

The actual change written as a fraction of the original is

$\frac{2}{10}$

The actual change is the numerator (top number).
The original number is the denominator (bottom number).

Convert the fraction to an equivalent fraction where the denominator is 100

$\frac{2}{10}=\frac{20}{100}=20\%$

OR

Write the actual change as a fraction of the original amount and multiply by 100.

$\frac{2}{10}\times100=20$

The percentage change of £12 from £10 is 20%

### Example 6: reverse percentages

The price of a coat is £40 after a 20% price cut.  Find the original price.

$100\%-20\%=80\%$

£40 represents 80% of the original price.

So 80% = £40

We need the original price which is 100%.

80% is 40

We can work out 10%

$40\div8=4$

Then we can work out 100%

$5\times 10 = 50$

OR

We can make an equation using x as the original number. Use the decimal multiplier and the new number.

\begin{aligned} x\times0.80&=40\\\\ x&=40\div0.80\\\\ x&=50 \end{aligned}

The price after the cut is £40. The original price was £50

### Common misconceptions

• Money needs two digits for the pence

E.g. Find 34% of £620

$620\times0.34=210.8$

• The decimal multiplier to work out a percentage increase can be greater than 1

E.g. To increase 50km by 3% we can work 103% of 50

$50\times1.03 =51.5$

• Percentages can be greater than 100%

E.g. Calculate the percentage change from 200 to 450.

The actual change is 450-200=250

$\frac{250}{200}\times100=125$

So the percentage change from 200 to 450 is 125%

### How to work out percentage practice questions

1. Find 40\% of \pounds 300 .

\pounds 1200

\pounds 120

\pounds 7.50

\pounds 12

10\% of \pounds 300 is \pounds 30 . We can multiply this by 4 to get 40\% .

2. Calculate 12.4\% of \pounds 3000 .

\pounds 3372

\pounds 37200

\pounds 372

\pounds 3720

As a multiplier, 12.4\% is 0.124 . To get the answer we can calculate 0.124\times3000

3. Increase 600m by 30\% .

630m

780m

180m

420m

30\% of 600 is 180 . We can add this on to the original amount to find the quantity after the increase.

4. Decrease 80 kg by 5\% .

75kg

95kg

76kg

84kg

5\% of 80   is  4 . We can subtract this from the original amount to find the quantity after the decrease.

5. Calculate the percentage change from 400 kg to 600 kg .

50\%

200\%

100\%

150\%

The actual increase is given by

600  −  400 = 200

The percentage increase is then given by

\frac{200}{400}\times100=50\%

6. A book costs \pounds 4 after a price reduction of 20\% . What was the original price?

\pounds 4.20

\pounds 5.00

\pounds 20.00

\pounds 4.25

\pounds 4 represents 80\% of the original amount, which means 10\% of the original amount is \pounds 0.50 .

Multiplying by ten to get 100\% means the original price was \pounds 5 .

### Percentages GCSE exam questions

1. Work out 90\% of \pounds 70.

(2 marks)

10\% is 7 , so 90\% is 9 × 7

(1)

9\times7 = 63

(1)

2. Charlotte invests \pounds 3000 for 4 years.

She gets a simple interest rate of 2\% per year.

Work out the total interest Charlotte gets.

(3 marks)

3000\times0.02=

(1)

4\times60

(1)

= \pounds 240

(1)

3. Last year Ron paid \pounds 450 for his car insurance.

This year he has to pay \pounds 603 for his car insurance.

Work out the percentage increase in his car insurance.

(3 marks)

The change is

603-450=

(1)

\frac{153}{450}\times 100

(1)

= 34\%

(1)

## Learning checklist

You have now learned how to:

• How to work out the percentage of a number
• How to calculate percentage change
• How to work out a percentage increase or decrease

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