Reverse Percentages: Finding the Original Amount (GCSE Maths)

A reverse percentage finds the original amount before a percentage increase or decrease was applied. Instead of multiplying by a multiplier, you divide by it. For example, if a price after a 20% increase is £60, the original price was £60 ÷ 1.2 = £50.

What is a reverse percentage?

A "forward" percentage question gives you the original and asks for the new amount. A reverse percentage question gives you the final amount and asks for the original. The key insight is that after a percentage change, the final amount is the original multiplied by a decimal multiplier — so to reverse it, you divide.

Situation | Multiplier | To reverse
20% increase | × 1.20 | ÷ 1.20
15% decrease | × 0.85 | ÷ 0.85
5% increase | × 1.05 | ÷ 1.05
30% decrease | × 0.70 | ÷ 0.70

How do you work out a reverse percentage step by step?

Follow this reliable three-step method:

Identify the multiplier. An increase of r% gives multiplier (100 + r) ÷ 100; a decrease of r% gives (100 − r) ÷ 100.

Divide the final amount by the multiplier. This gives the original.

Check. Apply the percentage change forwards to your original — you should arrive back at the final amount given.

Worked example: after a percentage increase

A jacket costs £78 after a 30% increase. What was the original price?

Multiplier for a 30% increase: 130 ÷ 100 = 1.3

Original = £78 ÷ 1.3 = £60

Check: £60 × 1.3 = £78. ✓

The original price was £60.

Worked example: after a percentage decrease

A television is on sale for £340 after a 15% reduction. What was the original price?

Multiplier for a 15% decrease: 85 ÷ 100 = 0.85

Original = £340 ÷ 0.85 = £400

Check: £400 × 0.85 = £340. ✓

The original price was £400.

What is the most common mistake with reverse percentages?

The most common error is working backwards by subtracting (or adding) the percentage of the final amount rather than dividing. For example, in the decrease question above, a wrong approach would be: £340 + 15% of £340 = £340 + £51 = £391. That is incorrect because 15% of £340 is not the same as 15% of the original £400. Always divide by the multiplier.

How do you handle VAT and price problems?

VAT at 20% is a very common real-life reverse-percentage context. If a price includes VAT and you need the price before VAT:

Multiplier for 20% VAT: 1.20

Price before VAT = final price ÷ 1.20

For example, a bill of £96 including 20% VAT: £96 ÷ 1.20 = £80 before VAT. The VAT itself is £96 − £80 = £16.

Why do reverse percentages appear so often at GCSE?

Reverse percentage questions test whether you understand what percentage change actually means rather than just applying a formula blindly. They appear on both Foundation and Higher papers and are worth two to four marks each. They also crop up in real life — sale prices, pay rises, and tax calculations all call for this skill.

Frequently asked questions

How do you do reverse percentages on a calculator?

Identify the multiplier (for example, 0.85 for a 15% decrease), then key in the final amount and divide by the multiplier. On most GCSE calculators: 340 ÷ 0.85 = gives 400. Always write the multiplier clearly in your working to gain method marks.

Why can't I just add the percentage back on?

Because percentages are calculated relative to the original, not the final amount. If the original is £400 and the price drops 15% to £340, then 15% of £340 is only £51, not £60. Adding £51 back gives £391, which is wrong. Dividing by the multiplier avoids this trap.

What if I am not told whether it is an increase or decrease?

Read the context carefully. Words like "reduced", "sale", "discount", and "fell" indicate a decrease (multiplier below 1). Words like "increased", "rose", "added VAT", and "marked up" indicate an increase (multiplier above 1). The question will always give enough information to identify which applies.

Can reverse percentages involve more than one change?

Yes — compound percentage changes. If a price rose 10% and then fell 5%, the combined multiplier is 1.10 × 0.95 = 1.045. To find the original from the final figure, divide by 1.045. Each step uses the same division-by-multiplier logic.

For Socratic percentage problem practice at GCSE, see aitutors.me.