To calculate a reverse percentage, divide the final amount by the multiplier that was applied to the original. If a price of £240 includes 20% VAT, the multiplier is 1.2, so the original price is £240 ÷ 1.2 = £200. If a sale price of £84 is after 30% off, the multiplier is 0.7, so the original was £84 ÷ 0.7 = £120. The common mistake is to take the percentage off the final figure instead, which gives the wrong answer.
Note: this is general information about the arithmetic. For tax, pricing or contracts, check the rules that apply, such as GOV.UK guidance on VAT, and the wording of any offer or agreement.
What a reverse percentage is
An ordinary percentage question starts with the original amount and asks for the new one: “increase £200 by 20%”. A reverse percentage starts with the new amount and asks for the original: “a price after 20% was added is £240, what was it before?”.
Reverse percentages appear in everyday UK situations:
- Taking VAT out of a price that includes it.
- Finding the full price of an item you only know the sale price of.
- Working out your salary before a pay rise, or your rent before an increase.
- Finding the pre-tax amount, the pre-markup cost or the amount before a service charge.
- Checking a claim such as “prices rose 8% to £54”.
The reason these questions cause trouble is that the percentage refers to the original amount, which you do not yet know. A 20% increase is 20% of the old number, not 20% of the new one. Once you see that, the fix is straightforward: work back through the same multiplier.
The multiplier method
Every percentage change can be written as a multiplier, which is the number you multiply the original by to get the new amount.
- To add a percentage, the multiplier is 1 + (percentage ÷ 100). Adding 20% gives 1.2, adding 5% gives 1.05 and adding 12.5% gives 1.125.
- To take off a percentage, the multiplier is 1 − (percentage ÷ 100). Taking off 30% gives 0.7, taking off 25% gives 0.75 and taking off 10% gives 0.9.
The relationship is: original × multiplier = new amount. To go backwards, you rearrange it:
original = new amount ÷ multiplier
That single line handles every reverse percentage question. The steps are:
- Decide whether the percentage was added or taken off.
- Write the multiplier (1.2, 0.7 and so on).
- Divide the amount you have by the multiplier.
- Check by multiplying the answer by the multiplier, to see that you get back to the amount you started with.
Here are the three examples from the introduction, computed with the House of Calculator reverse percentage tool:
| What you know | Multiplier | Working | Original |
|---|---|---|---|
| £240 including 20% VAT | × 1.2 | £240 ÷ 1.2 | £200 |
| £84 after 30% off | × 0.7 | £84 ÷ 0.7 | £120 |
| £31,500 after a 5% pay rise | × 1.05 | £31,500 ÷ 1.05 | £30,000 |
Each answer can be verified. £200 × 1.2 = £240, £120 × 0.7 = £84 and £30,000 × 1.05 = £31,500. The check takes seconds and catches most errors.
Why taking the percentage off the final number is wrong
The tempting shortcut is to take 20% off £240, which gives £192, or to add 30% to £84, which gives £109.20. Both are wrong, because each percentage is being applied to the wrong base.
For the VAT example, 20% of £200 (the original) is £40. But 20% of £240 (the final price) is £48, which is more than the VAT actually in the price. Taking £48 off gives £192, and the true VAT-exclusive price is £200, so the shortcut is £8 too low.
For the sale example, 30% was taken off £120 (the original), and that is £36. 30% of £84 is £25.20, and adding it to £84 gives £109.20, which is £10.80 below the true £120.
This table shows how large the error can be in each example:
| Situation | Correct original | Wrong shortcut | Error |
|---|---|---|---|
| £240 including 20% VAT | £200 | £192 (20% off £240) | £8 too low |
| £84 after 30% off | £120 | £109.20 (30% on £84) | £10.80 too low |
| £31,500 after a 5% rise | £30,000 | £29,925 (5% off £31,500) | £75 too low |
The error is always an understatement when you reverse a percentage the wrong way, and it grows with the size of the percentage. At 5% the mistake is small, at 30% it is large, and at 50% off the shortcut is badly wrong: £50 after 50% off was £100, but adding 50% to £50 gives £75. The VAT guides in this series, such as VAT explained: adding and removing VAT, use the same divide-by-1.2 rule for the same reason.
Worked examples from everyday life
Taking VAT out of a price. At the 20% standard rate, divide the VAT-inclusive price by 1.2. A VAT-inclusive price of £69.30 at a 10% rate would be £63 net, but at 20% it is £57.75. The VAT is the difference. For a price that includes the 5% reduced rate, such as £63, you divide by 1.05 to get £60, so £3 of VAT. For a business, VAT for small businesses: registration and what to charge explains when you need to charge VAT and how to reclaim it.
Finding the original price in a sale. A coat is £45 in a sale marked “25% off”. The multiplier is 0.75, so the original was £45 ÷ 0.75 = £60, and the saving was £15. If a shop says “was £70” and the sale price is £45, the true discount is (70 − 45) ÷ 70 = 35.7%, not 25%, so check any claim you doubt.
Working out a salary before a rise. After a 5% pay rise your salary is £31,500. Divide by 1.05 to get the old salary of £30,000, so the rise was £1,500. If instead you wanted to know what 5% of the new salary is, that is £1,575, which would give the wrong increase.
Reversing a cut. A car loses 40% of its value and is now worth £18,000. The multiplier is 0.6, so the original value was £18,000 ÷ 0.6 = £30,000, and the loss was £12,000.
Taking off a service charge. A restaurant bill of £112.50 includes a 12.5% service charge. Divide by 1.125 to find the food and drink total of £100, and the service charge is £12.50. Check the bill itself, because the charge may be listed separately.
Reversing a mark-up. A product sells for £150 after a 25% mark-up on cost. Divide by 1.25 to find the cost of £120. Mark-up is based on cost, whereas margin is based on the selling price, so the two are not the same, a point covered in our Break-even point and profit margin: how to work them out guide.
Reversing two or more changes in a row
Sometimes a price has gone through more than one change, such as two successive discounts, or a rise followed by a fall. The multipliers multiply together. To go back, divide by the combined multiplier.
Suppose an item is reduced by 15%, and then a further 10% is taken off at the till, and you pay £76.50. The two multipliers are 0.85 and 0.9, so the combined multiplier is 0.85 × 0.9 = 0.765. The original price was £76.50 ÷ 0.765 = £100. The overall discount is 23.5%, not 25%, because the second 10% is taken off the already reduced price.
Another trap is a rise followed by an equal fall. If £100 goes up 20% to £120 and then down 20%, the result is £96, not £100, because the 20% fall applies to the larger number. The combined multiplier is 1.2 × 0.8 = 0.96. To reverse the pair from £96, divide by 0.96 and get back to £100. The same logic explains why a series of changes does not simply cancel out, and Percentage increase and decrease, and reverse percentages shows how the compound effect builds up.
When you are asked to reverse more than one change, it helps to write each multiplier in order, multiply them, and divide once, instead of dividing step by step. The result is the same, but there is less to go wrong.
Checking your answer and spotting unreasonable results
A few quick checks save time:
- Direction. If a percentage was added, the original must be smaller than the final amount. If a percentage was taken off, the original must be larger.
- Size. For small percentages, the original should be close to the final amount. For 5%, it is roughly 95% of the final amount.
- Forward test. Apply the percentage to your answer. If you do not return to the amount you began with, something is wrong.
- Rounding. Prices and wages are rounded to pence, so a reverse percentage can give a figure like £62.999 that you round to £63.00. Round at the end, not in the middle.
It also helps to estimate. For 20%, dividing by 1.2 is the same as taking away one sixth of the amount: £240 less £40 (a sixth) gives £200. For 25% off, the original is the final amount plus a third: £45 plus £15 gives £60. These shortcuts are handy for mental maths, but use the calculator for anything you will rely on.
Common mistakes with reverse percentages
- Applying the percentage to the final amount. The percentage belongs to the original, so divide by the multiplier instead.
- Using the wrong multiplier. Adding 20% is 1.2 and taking off 20% is 0.8. They are not interchangeable.
- Subtracting the percentage points. A rise of 20% followed by a fall of 20% is not a return to the start.
- Forgetting that discounts and VAT combine. A price with a discount and VAT has two multipliers, and the order of the steps can matter in practice.
- Confusing mark-up and margin. A 25% mark-up is a 20% margin, so the base is different.
- Rounding too early. Keep the full figure until the last step.
- Skipping the check. Multiplying back takes a few seconds and shows whether the answer is right.
Try the calculator
Enter the amount you have, say whether a percentage was added or taken off, and pick or type the percentage in the Reverse Percentage and VAT-Inclusive Price Calculator to get the original, the amount added or taken off, and the multiplier used.
Frequently asked questions
How do I find the original price before VAT?
Divide the VAT-inclusive price by 1.2 for the standard 20% rate, or by 1.05 for the 5% reduced rate. For £240 including 20% VAT, the net price is £200 and the VAT is £40.
How do you work out the original price after a discount?
Divide the sale price by 1 minus the discount as a decimal. For 30% off, divide by 0.7. A sale price of £84 gives an original of £120.
Why can't I just add the percentage back on?
Because the percentage was taken from the original amount, not from the reduced one. Adding 30% to £84 gives £109.20, but the original was £120, since 30% of £120 is £36 and £120 minus £36 is £84.
What is the formula for a reverse percentage?
Original = final amount ÷ multiplier, where the multiplier is 1 plus the percentage for an increase or 1 minus the percentage for a decrease, both as decimals.
Can I reverse two percentage changes at once?
Yes. Multiply the two multipliers together and divide the final amount by the result. For 15% off then 10% off, the combined multiplier is 0.765, so £76.50 was originally £100.