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Percentages Percentage increase Percentage decreaseProportion

This topic is relevant for:

Here we will learn about **reverse percentages **including how to work backwards to find an original amount given a percentage of that amount or a percentage increase/decrease.

There are also reverse percentages worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

**Reverse percentages **(or inverse percentages) means working backwards to find an original amount, given a percentage of that amount.

- We can do this using a calculator by taking the percentage we have been given, dividing to find
1% and then multiplying by100 to find100% . - We can also do this without a calculator by using factors of the percentage we have been given.
- Sometimes we are given a percentage of an amount and we need to work out what the original value was.

We need to remember that the **original amount is 100% **of the value.

In order to find the original amount given a percentage of the amount (using a calculator):

**Write down the percentage and put it equal to the amount you have been given.****Divide both sides by the percentage.**

(e.g. if you have80% , divide both by80 ). This will give you1% .**Multiply both sides by**100 .

This will give you100% .

Get your free reverse percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free reverse percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**Put the percentage equal to the amount.**

2 **Divide both sides by the percentage to find 1%. **

In this case the percentage is 45%, so divide by 45.

3 **Multiply by 100 to find 100%.**

The original number was

**Put the percentage equal to the amount.**

**Multiply by 100 to find 100%.**

The original number was

In a situation where we do not have a calculator, we can often simplify the problem by using common factors.

Rather than finding

**Write down the percentage and put it equal to the amount you have been given.****Identify a common factor of the percentage and**100% (a number which goes in to both).**Use division to find that percentage of your amount.****Use multiplication to find**100% .

**Put the percentage equal to the amount.**

**Identify a common factor of 70% and 100%.**

Factors of **1**, **2**, **5**, 7, **10**, 14, 35, 70

Factors of **1**, **2**, 4, **5**, **10**, 20, 25, 50, 100

Here

**As we now have 10%, we need to multiply by 10 to find 100%.**

The original amount was

Note: In this example, and every example, the method of finding

**Put the percentage equal to the amount.**

**Identify a common factor of 125% and 100%.**

Factors of **5**, **25**, 125

Factors of ** 5**, 10, 20, **25**, 50, 100

Here we are going to use

**As we now have 25%, we need to multiply by 4 to find 100%.**

The original amount was

Sometimes, instead of being told a percentage of the amount, we are told what percentage increase or decrease has occurred.

The only difference here compared to what we have already looked at is that we first need to identify what percentage of the original amount we now have.

**Identify what percentage of the original amount you now have.**

If it has been increased by a percentage, add that percentage onto100% .**If it has been decreased by a percentage, subtract that percentage from**100% .**Write down the percentage and put it equal to the amount you have been given.****Use either the calculator or non-calculator method to find**100% .

The number of fans attending a football match this week was 12% more than last week. If 728 people attended the match this week, how many attended last week?

**This is a percentage increase of 12%.**

**Write down the percentage and put it equal to the amount you have been given.**

**This is a calculator question, so use the method of finding 1%.**

The number of fans last week was 650.

The value of a car has decreased by

**This is a percentage decrease of 16.5%. **

**Write down the percentage and put it equal to the amount you have been given.**

**Use a calculator to find 1%.**

The original price was

A puppy’s weight has increased by

**This is a percentage increase of 20%.**

**Write down the percentage and put it equal to the amount you have been given.**

**This is a non-calculator question, so use the common factor method.**

Factors of **20**, 24, 30, 40, 60, 120

Factors of **20**, 25, 50, 100

There are several common factors here. It would be easiest to use

The puppy’s original weight was

A television is in a

**This is a percentage decrease of 10%.**

**Write down the percentage and put it equal to the amount you have been given.**

**This is a non-calculator question, so use the common factor method.**

Factors of **10**, 15, 30, 45, 90

Factors of **10**, 20, 25, 50, 100

Here we are going to use

The original price of the television was

**Calculating a percentage and adding it on**

A common mistake is to work out the percentage of the number and then add it on.

E.g.

Given

Remember, this does not work as

**Not adding/subtracting from**100% when it is a percentage increase/decrease

A common mistake is to use the actual percentage increase/decrease rather than adding/subtracting from

E.g

If you are told it is a

Reverse percentages is part of our series of lessons to support revision on percentages. You may find it helpful to start with the main percentages lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

You may use a calculator for questions 1, 2 and 5.

1. 36\% of a number is 324 . Find the original number.

117

900

360

1166

36\%=324

Dividing both sides by 36 gives

1\%=9

Multiplying both sides by 100 gives

100\%=900

2. 145\% of a number is 2900 . Find the original number.

3045

2755

2000

1885

145\%=2900

Dividing both sides by 145 gives

1\%=20

Multiplying both sides by 100 gives

100\%=2000

3. 60\% of a number is 210 . Find the original number.

126

525

270

350

60\%=210

Dividing both sides by 3 gives

20\%=70

Multiplying both sides by 5 gives

100\%=350

4. 150\% of a number is 33 . Find the original number.

44

22

49.5

11

150\%=33

Dividing both sides by 3 gives

50\%=11

Multiplying both sides by 2 gives

100\%=22

5. The price of a car is reduced by 15\% . The reduced price is £6800 . Find the original price.

\pounds 8000

\pounds 7820

\pounds 5780

\pounds 6785

The price has been reduced by 15\% , so \pounds 6800 is equal to 85\% of the original price.

85\%=6800

Dividing both sides by 85 gives

1\%=80

Multiplying both sides by 100 gives

100\%=8000

6. The number of customers who visited a shop today was 10\% higher than the number who visited yesterday. Today 231 customers visited the shop. How many customers visited the shop yesterday?

221

210

241

218

The number of customers has increased by 10\% , so 231 is equal to 110\% of the number from yesterday.

110\%=231

Dividing both sides by 11 gives

10\%=21

Multiplying both sides by 10 gives

100\%=210

1. 40\% of the children in Rahim’s class walk to school.

12 children walk to school.

How many children are in Rahim’s class?

**(2 marks)**

Show answer

40\% = 12

** (1)**

10\% = 3

** (1)**

100\% = 30

** (1)**

2. In a sale, prices are reduced by 15\% .

A phone is reduced by \pounds 36 .

Find the original price of the phone.

**(3 marks)**

Show answer

15\% = \pounds 36

** (1)**

5\% = \pounds 12

** (1)**

100\% = \pounds 240

** (1)**

3. Tony receives a pay increase of 12\% .

His new salary is \pounds 31920 per annum.

Calculate how much more money he earns each year following the pay increase.

**(4 marks)**

Show answer

112\% = \pounds 31920

** (1)**

1\% = \pounds 285

** (1)**

100\% = \pounds 28500

** (1)**

\pounds 31920 − \pounds 28500 = \pounds 3420

** (1)**

You have now learned how to:

- Find an amount given a percentage of that amount (calculator and non-calculator)
- Calculate reverse percentages involving percentage increase/decrease

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