To do a reverse percentage, work out what percentage of the original the final figure represents, divide the final figure by that percentage to find 1%, then multiply by 100. After a 20% discount you are paying 80% of the original price, so a sale price of £160 means 1% is £160 ÷ 80 = £2 and the original price was £200. The quicker version is to divide by the decimal multiplier: £160 ÷ 0.8 = £200.
Reverse percentages come up often in employer tests, usually as discounted prices, figures including VAT or a change in headcount. You can practise them in a free numerical reasoning test, where they sit inside tables of business data.
What is a reverse percentage?
A reverse percentage calculation works backwards. You know the figure after a percentage increase or decrease has been applied, and you need the original figure before the change. It is sometimes called back-calculating a percentage.
The question usually gives itself away with words such as “before the discount”, “excluding VAT”, “last year” or “originally”.
The reverse percentage formula
Original value = final value ÷ percentage the final value represents × 100
Or, using the decimal multiplier:
Original value = final value ÷ multiplier
The only step that needs thought is the percentage the final value represents:
- After a decrease of x%, the final value is (100 − x)% of the original. After a 20% fall, it is 80%, so the multiplier is 0.8.
- After an increase of x%, the final value is (100 + x)% of the original. After a 20% rise, it is 120%, so the multiplier is 1.2.
How to calculate a reverse percentage, step by step
Take an item that costs £160 in a 20% off sale.
- Start at 100% and apply the change. A 20% reduction means you pay 100% − 20% = 80% of the original price. So £160 is 80%.
- Find 1%. Divide the final figure by the percentage it represents: £160 ÷ 80 = £2.
- Multiply by 100. £2 × 100 = £200. The original price was £200.
- Check by working forwards. 20% of £200 is £40, and £200 − £40 = £160, which matches the question.
The 1% method is easiest without a calculator. With one, dividing by the multiplier is quicker: £160 ÷ 0.8 = £200.
The mistake to avoid
The most common error is to take the percentage of the final figure and add it back on. Adding 20% to £160 gives £160 × 1.2 = £192, not £200.
It fails because the 20% was taken from the original, larger price, not from the sale price. Any time you see the answer you would get by reversing the percentage on the final figure, treat it as a trap. In multiple choice tests it is almost always one of the options.
Worked examples
Example 1: original price after a discount
A scarf is marked 75% off and now costs £20. What was the original price?
- After 75% off, the price is 100% − 75% = 25% of the original.
- Find 1%: £20 ÷ 25 = £0.80.
- Multiply by 100: £0.80 × 100 = £80.
The original price was £80. Check: 25% of £80 is £20.
Example 2: original price after an increase
A coffee shop raised its prices by 10%, and an Americano now costs £2.75. What did it cost before the rise?
- After a 10% rise, the new price is 100% + 10% = 110% of the old one.
- Find 1%: £2.75 ÷ 110 = £0.025.
- Multiply by 100: £0.025 × 100 = £2.50.
The Americano cost £2.50 before the rise. Check: £2.50 × 1.1 = £2.75.
Example 3: taking VAT off a price
A laptop costs £660 including VAT at 20%. What is the price before VAT?
- Including VAT, the price is 120% of the net price, so the multiplier is 1.2.
- Divide by the multiplier: £660 ÷ 1.2 = £550.
The price before VAT is £550. Check: £550 × 1.2 = £660. Note that the VAT itself is £660 − £550 = £110, which is not the same as 20% of £660 (£132).
Example 4: a fall in headcount
A company’s headcount fell by 15% over the year to 2,040 employees. How many did it have at the start of the year?
- After a 15% fall, 2,040 is 100% − 15% = 85% of the original.
- Find 1%: 2,040 ÷ 85 = 24.
- Multiply by 100: 24 × 100 = 2,400.
The company started the year with 2,400 employees. Adding 15% to 2,040 would give 2,346, which is the wrong answer test writers expect to see.
Example 5: two changes in a row
A product’s price was raised by 20% and then cut by 10% in a sale. It now sells for £54. What was the price before either change?
- Combine the two multipliers: 1.2 × 0.9 = 1.08. The two changes together are the same as a single 8% rise.
- Divide by the combined multiplier: £54 ÷ 1.08 = £50.
The original price was £50. Check: £50 × 1.2 = £60, and £60 × 0.9 = £54.
Do not add and subtract the percentages (20% − 10% = 10%) and divide by 1.1. That gives £49.09, because the 10% cut was applied to the higher price, not the original.
Quick reference
| Change | Final figure is | Divide by |
|---|---|---|
| 10% increase | 110% of original | 1.1 |
| 20% increase (for example VAT) | 120% of original | 1.2 |
| 5% decrease | 95% of original | 0.95 |
| 15% decrease | 85% of original | 0.85 |
| 25% decrease | 75% of original | 0.75 |
If you need to work out the size of a change rather than undo one, see our guides to calculating percentage increase and calculating percentage decrease.
Practise reverse percentages
In an employer test, reverse percentages rarely arrive as a bare sum. They come wrapped in a sales table or a price list, and you need to spot that the question is asking for an earlier figure. Our numerical reasoning test examples include one worked through in that format. Then build speed with a timed numerical reasoning practice test.
