# Reverse Percentages

GCSELevel 6-7Cambridge iGCSE

## Reverse Percentages

Finding the original value before a percentage change has been applied is called a reverse percentage.

Once you are able to recognise it is a reverse percentage question, the method is very simple.

## Method 1: Calculating $1\%$

1. Write the amount given as a percentage of the original amount.
2. Find $1\%$ of the original amount by dividing by this value
3. Multiply by $\textcolor{#aa57ff}{100}$

Example:

The price of a pair of trainers has been decreased by $20\%$.

If its new price is $£60.80$, work out its original price.

1. The trainers have been decreased by $20\%$, so $£60.80$ represents $\textcolor{#10a6f3}{80\%}$ of its original value.
2. Divide by $80$ to find $1\%$ of the original price.
3. Multiply by $\textcolor{#10a6f3}{100}$

$£60.80 \div80=£0.76$ $(1\%)$

$£0.76\times 100=\textcolor{#10a6f3}{£76}$ (original price)

Level 6-7GCSECambridge iGCSE

## Method 2: Using Decimals

If you are comfortable with the method described above, you may find it easier to skip step $2$ and $3$.

This can be done by converting the percentage found in step $1$ into a decimal and dividing by this value instead.

Example:

The value of a painting has increased by $30\%$ over the course of a year to $£858$.

Work out the value of the painting a year ago.

1. $£858$ represents $\textcolor{#10a6f3}{130\%}$ of the original value of the painting.
2. Convert this percentage to a decimal: $130\%=\textcolor{#10a6f3}{1.3}$
3. Divide by $1.3$

$£858 \div 1.3=\textcolor{#10a6f3}{£660}$

Level 6-7GCSECambridge iGCSE

## Example

As part of a black Friday deal a laptop has been reduced by $\textcolor{#10a6f3}{28\%}$ to $£547.20$.

Calculate the original price of the laptop before the discount.

[2 marks]

First find what percentage of the original price $£547.20$ is.

$100\%-28\%=\textcolor{#10a6f3}{72\%}$

Convert to a decimal:

$72\div 100=\textcolor{#10a6f3}{0.72}$

Divide the new price by this decimal:

$£547.20 \div 0.72=\textcolor{#10a6f3}{£760}$

Level 6-7GCSECambridge iGCSE

## Reverse Percentages Example Questions

Question 1: The value of a car depreciates by $30\%$ one year after Erin bought it.

If the value of the car is now $£5880$, how much did she buy the car for?

[2 marks]

Level 6-7GCSE Cambridge iGCSE

$£5880$ is $70\%$ of the original value of the car.

Original value $=£5880\div 0.7=£8400$

Gold Standard Education

Question 2: The population of a town increases by $3.4\%$ every year.

The town’s population was $55319$ in 2021, what was the population of the town in 2020?

[2 marks]

Level 6-7GCSE Cambridge iGCSE

$55319$ represents $103.4\%$ of the population of the town in 2020.

Population of the town in 2020 $=55319 \div 1.034=53500$

Gold Standard Education

Question 3: A striker for a football team scores $27$ goals in one season, which is $30\%$ of the team’s total goals for that season.

Calculate the total number of goals scored by the football team in that season.

[2 marks]

Level 6-7GCSE Cambridge iGCSE

$27$ goals represents $30\%$ of the team’s total goals.

Total number of goals $=27\div 0.3=90$