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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
Percent means “number of parts per hundred” and the symbol we use for percent is the percent sign
E.g.
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
This is the same as finding a
The answer is
Step-by-step guide: Percentage of an amount
- Percentage conversions
Percentages, fractions and decimals can all be used to represent part of a whole.
For example,
It is useful to be able to convert between percentages, fractions and decimals.
To change a percentage into a fraction, write the percentage number as the numerator of a fraction and 100 as the denominator, and then simplify the resulting fraction if possible.
E.g.
42\%= \frac{42}{100} = \frac{21}{50}To change a fraction into a percentage, there are two methods.
First see if it is possible to write an equivalent fraction with a denominator of 100; then the numerator will be the percentage.
Eg. \frac{7}{20} = \frac{7\times5}{20\times5}=\frac{35}{100}=35\%
A second method is to carry out the division represented in the fraction and then multiply by 100.
E.g.
\frac{1}{8} = 1 \div 8 =0.125 0.125 \times 100 = 12.5 \frac{1}{8} = 12.5 \%To change a percentage into a decimal, divide the percentage number by 100.
Eg.
73\% = 73 \div 100 = 0.73
To change a decimal into a percentage, multiply the decimal by 100.
E.g. 0.29
0.29 \times 100 = 29 29\%- 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
We can use
The answer is
Step-by-step guide: Percentage multipliers
- Percentage as operator
In order to calculate a percentage of an amount, a percentage increase or a percentage decrease we can use a percentage multiplier. To do this we change the percentage that we want into a decimal, and then multiply the amount by that decimal to calculate the answer.
For example,
34% of 58.
Here we want 34% which as a decimal is 0.34
Therefore the calculation is
58 \times 0.34 = 19.72For example,
Increase 78 by 15%
Here we want an increase of 15% which means in total we want 115%, which as a decimal is 1.15
Therefore the calculation is
78 \times 1.15 = 89.7For example,
Decrease 45 by 20%
Here we want a decrease of 20% which means in total we want 80%, which as a decimal is 0.8
Therefore the calculation is 45 \times 0.8 = 36
- Percentage increase
This is where we are asked to increase (make bigger) a value by a certain amount.
E.g. Increase
We can find
The answer is
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
We can find
The answer is
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
The actual change from
The answer is
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.
To find the original number we need to find
The answer is
Step-by-step guide: Reverse percentages
- Percentage calculations
Let’s look at different methods that can be used to perform percentage calculations.
For example,
What is 40% of 70?
Method 1: The one percent method
Find 1% first by dividing the amount by 100 and then multiply the amount by the percent you want.
\frac{70}{100}\times 40 = 28Method 2: The decimal multiplier method
Write the percent you want as a decimal and then multiply the amount by this decimal.
40\%=0.4 70 \times 0.4=28Method 3: Using equivalent fractions
Write the percent you want as a fraction in simplest form and then multiply the amount by this fraction.
40\% = \frac{40}{100} =\frac{4}{10} =\frac{2}{5} 70 \times \frac{2}{5} = \frac{70\times 2}{5} = \frac{140}{5} = 28Method 4: Building up an answer from simple percentages you know
Using simple percentages you can “
build up the answer to the question.
For this question if you know 10% of 70 then you can multiply this answer by 4 to find 40%.
10\% of 70 = 7 40\% of 70 = 7 \times 4 = 28Note, methods 1 and 2 lend themselves best to questions where you are allowed to use a calculator. Methods 3 and 4 are most helpful when calculators are not permitted.
See also: One number as a percentage of another
What are the 6 different types of percentage questions?
How to work out percentage worksheet
Get your free how to work out percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREE How to work out percentage worksheet
Get your free how to work out percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.
DOWNLOAD FREEEquation to percentage
We could be asked to solve problems where we need to form and solve equations to find answers as percentages. We often see this when dealing with interest and depreciation calculations.
For example,
A car purchased for £36000 depreciates by x% each year. If after 3 years, the car has a value of £15187.50, what percentage does the car depreciate by annually?
If we were solving this as a depreciation question with the percentage value known and the car value unknown we would use the calculation
36000 \times \text { multiplier }^{3}=15187.50We need to work backwards to find the multiplier. Essentially solving the equation, for ease let’s call the multiplier, x.
36000 \times x^{3}=15187.50 x^{3}=\frac{15187.50}{36000} x^{3}=0.421875 x=\sqrt[3]{0.421875} x=0.75The depreciation multiplier used for this calculation was 0.75, this tells us that the final car value of £15187.50 was 75% of its original value. 100% – 75% = 25%, so the value of the car depreciated by 25% each year.
Step-by-step guide: Percentage profit
How to find the percentage
Percentages examples
Example 1: percentage of an amount
Find
- Write down what you have and what you are trying to find.
We need to work out
2 Work out what you need.
3 Write down the final answer.
Example 2: percentage multipliers
Find
Write down what you have and what you are trying to find.
We want to find
Work out what you need.
We can write the percentage as a decimal number.
This gives the percentage in decimal form which is the percentage multiplier.
Write down the final answer.
Example 3: percentage increase
Increase
Write down what you have and what you are trying to find.
We want to find
Work out what you need.
So
OR
You could also use the decimal multiplier.
Write down the final answer.
Example 4: percentage decrease
Decrease
Write down what you have and what you are trying to find.
We want to find
Work out what you need.
So
OR
You could use the decimal multiplier.
Write down the final answer.
Example 5: percentage change
The price of a t-shirt increased from
Write down what you have and what you are trying to find.
The original amount is
We need to find the change as a percentage of the original amount.
Work out what you need.
The actual change is
The actual change written as a fraction of the original is
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
OR
Write the actual change as a fraction of the original amount and multiply by
Write down the final answer.
The percentage change of
Example 6: reverse percentages
The price of a coat is
Write down what you have and what you are trying to find.
£40 represents 80% of the original price.
So
We need the original price which is
Work out what you need.
We can work out
Then we can work out
OR
We can make an equation using
Write down the final answer.
The price after the cut is
Common misconceptions
- Money needs two digits for the pence
E.g. Find
The answer is
- The decimal multiplier to work out a percentage increase can be greater than
1
E.g. To increase
The answer is
- Percentages can be greater than
100%
E.g. Calculate the percentage change from
The actual change is
So the percentage change from
How to work out percentage practice questions
1. Find 40\% of \pounds 300 .
10\% of \pounds 300 is \pounds 30 . We can multiply this by 4 to get 40\% .
2. Calculate 12.4\% of \pounds 3000 .
As a multiplier, 12.4\% is 0.124 . To get the answer we can calculate 0.124\times3000
3. Increase 600m by 30\% .
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\% .
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 .
The actual increase is given by
600 − 400 = 200The 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 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)
(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
The next lessons are
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