Reverse percentages

Reverse percentages help us to work out the original price or value of an item after it has been increased or decreased in value, for example, following a price increase or a sale.

Calculating reverse percentages

Calculating reverse percentages depends on knowing that before an increase or decrease in price, an item is always worth 100% of its value, no matter what that value is. This is because 100% represents the whole amount or the full price.

Example

A shop has a sale where 20% is taken off all prices. A top is now worth £24. What price was it originally?

A common mistake is to work out 20% of £24 and add this on to £24. This will not work as 20% of £24 is not as much proportionally as 20% of the bigger, original amount.

The original price of the top is unknown, but no matter what this price was, this is 100% of the value. The shop has then reduced prices by 20%. This means that 80% of the value of the top remains ( 100 \% - 20 \% = 80 \%) and this is worth £24.

To find the original price of the item, 100% has to be found. There are many ways to do this, but using a unitary method is a method that will always work.

80% = 24

Divide both sides by 80 to get 1%: 1 \% = 24 \div 80 = 0.3

Multiply both sides by 100 to get 100%: 100 \% = 0.3 \times 100 = 30

100% of the value of the top is worth £30 which means before the sale of 20%, the top cost £30.

This answer can be tested by taking 20% off £30. If the answer is £24, then the method and answer are correct.

\pounds 30 - 20 \% = \pounds 30 - \pounds 6 = \pounds 24

Question

An antique is sold for £550 which is a 10% increase on the price that it was originally bought for. How much was the antique originally bought for?

The original price of the antique is unknown, but no matter what this price was, this is 100% of the value. The item has then been sold for 10% more. This means that it’s now 110% and this is worth £550.

110 \% = 550

Divide both sides by 110 to get 1%:

1 \% = 550 \div 110 = 5

Multiply both sides by 100 to get 100%:

100 \% = 5 \times 100 = \pounds 500