Reverse percentages

A reverse percentage problem gives the amount after a change and asks for the original.

1. Write the new amount as a percentage of the original (for example, 120% after a 20% increase).
2. Divide the new amount by that percentage as a multiplier.

Example: after a 20% increase, a price is £84. What was it before?

1. £84 is 120% of the original, so the multiplier is 1.2
2. 84 ÷ 1.2 = £70

Check: 70 × 1.2 = 84.

The method is the same: the new amount is less than 100% of the original.

Example: in a 30% off sale, a coat costs £56. What was the original price?

1. £56 is 100% − 30% = 70% of the original, so the multiplier is 0.7
2. 56 ÷ 0.7 = £80

Example: a TV costs £150 including 20% VAT. What is the price before VAT?

1. £150 is 120% of the price before VAT
2. 150 ÷ 1.2 = £125

Check: 125 × 1.2 = 150.

You can also find 1% first, then 100%.

1. Write which percentage the amount you know is.
2. Divide to find 1%.
3. Multiply by 100 to find the original.

Example: after a 20% increase, a price is £84. What was it before?

1. 120% = £84
2. 1% = 84 ÷ 120 = £0.70
3. 100% = 0.70 × 100 = £70

This gives the same answer as dividing by the multiplier, and it helps when you can't use a calculator.