Reverse Percentages | Questions and Worksheets | MME

Reverse Percentages – Worksheets and revision

Level 4-5
GCSE Maths Revision Guide

Reverse Percentages

Reverse percentages are where we are given a value or an amount that has increased or decreased by a certain percent, and then we have to use this information to calculate the original amount. 

Make sure you are happy with the following topics before continuing.

Method 1: Reverse Percentages 1% to 100%

  • Write the amount in the question as a percentage of the original value
    • i.e. write an increase of 10\% as 110\% and a decrease of 10\% as 90\%
  • Divide the amount by this value to find 1\% of the original value
  • Multiply by 100  

Example:

Felicity buys a dress in a sale. It is reduced by \textcolor{red}{40\%} down to a price of \textcolor{blue}{£54}. Work out the original price of the dress.

Step 1: We need to work out the cost of the dress as a percentage of its original value. We know it was \textcolor{red}{40\%} off so:

100\% - \textcolor{red}{40\%} = 60\%

Step 2: Divide the cost by 60\% to find 1\% of the original value

\begin{aligned} (\div \, 60) \, \, \, \, \, 60\% &= \textcolor{blue}{£54} \,\,\,\,\, (\div \, 60) \\ 1\% &= £0.9 \end{aligned}

Step 3: Multiply by 100 to get 100\% and the original value

\begin{aligned} (\times \, 100) \, \, \, \, \, 1\% &= £0.9 \,\,\,\,\, (\times \, 100) \\ 100\% &= \textcolor{purple}{£90} \end{aligned}

Level 4-5

Method 2: Reverse Percentages using decimals

  • Write the amount in the question as a percentage of the original value
    • i.e. write an increase of 10\% as 110\% and a decrease of 10\% as 90\%
  • Convert this percentage to a decimal or a fraction.
  • Divide the amount by this value to find the original value

Example:

Felicity buys another dress in a sale. It is reduced by \textcolor{red}{30\%} down to a price of \textcolor{blue}{£42}. Work out the original price of the dress.

Step 1: We need to work out the cost of the dress as a percentage of its original value. We know it was \textcolor{red}{30\%} off so:

100\% - \textcolor{red}{30\%} = 70\%

Step 2: Convert this percentage to a decimal or fraction 70\% = 0.7 

Step 3: Divide the sale value by this decimal equivalent to get the original value

£42 \div 0.7 = \textcolor{purple}{£60}

Level 4-5

Example 1: Reverse Percentage

A TV cost \textcolor{limegreen}{£160} in a \textcolor{blue}{20\%} off sale. Calculate the original price of the TV before the sale. 

[1 mark]

Step 1: 80\% = \textcolor{limegreen}{£160}

Step 2: Divide both sides by 80 to find 1\% = £2

Step 3: Multiply by 100 to find 100\%, i.e. the original amount: 100\% = \textcolor{purple}{£200}

Level 4-5

Example 2: Reverse Percentage

A gym’s membership increased in price by \textcolor{red}{15\%} to \textcolor{orange}{£23} per month. What was the cost of the membership before the increase? 

[1 mark]

Step 1: 115\% = \textcolor{orange}{£23}

Step 2: Divide both sides by 115 to find 1\% = £0.20

Step 3: Multiply by 100 to find 100\%, i.e. the original amount: 100\% = \textcolor{purple}{£20} per month

Level 4-5

Example 3: Reverse Percentage

Rogelio’s new record for the 100m sprint is \textcolor{red}{10.8} seconds. This is \textcolor{limegreen}{5.8\%} faster than his previous record. What was his previous record (to 2 dp)? 

[1 mark]

Step 1: 94.2 \% = \textcolor{red}{10.8} seconds

Step 2: Divide both sides by 94.2 to find 1\% = 0.11464

Step 3: Multiply by 100 to find 100\%, i.e. the original amount: 100\% = \textcolor{purple}{11.46} seconds (2 dp)

Level 4-5
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Example Questions

To find the original price of the t-shirt we can divide the new cost by 0.75:

 

£13.50\div0.75=£18

To find the original cost of the car we can divide the new cost by 1.05:

 

£15,550\div1.05=£14,761.90

To find the total mass of the bar we can divide the mass of protein by 0.18:

 

15.6g \div \, 0.18 = 86.67g

To find the total capacity of the stadium we can divide the recent attendance by 0.85:

 

46,235\div 0.85 =54,394

To find the total population of the U.K. we can divide the population of London by 0.14:

 

9,300,000\div 0.14 =66,428,571

If a car has decreased in value by 56\%, this means that the car is now worth 44\% of what it was worth before (100\% - 56\% = 44\%).

This means that the current value of £10,120 represents 44\% of the original price.

The original price is the 100\% amount, so we need to work out what 100\% represents if 44\% = £10,120. The easiest way to do this is to work out what 1\% is:

If

44\% = \pounds10,120

then

1\% = £10,120 \div 44 = \pounds230

 

If

1\% = \pounds230

then

100\% = £230 \times100 = \pounds23,000

 

So, the original price of Petra’s car was £23,000.

 

Another way we can view this question is by working what we would multiply the original amount by in order to work out the new value. If we are calculating a 56\% decrease, we would multiply the original amount by 0.44 to work out the new value. Therefore, if we know the new amount and wish to work out the original amount, then we can simply divide the new amount by 0.44:

\pounds10,120 \div 0.44 = \pounds23,000

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