Reverse Percentage Calculator
Find the original value before a percentage increase or discount.
The reverse percentage formula
Original value = Final value ÷ (1 ± percentage/100)
Use minus when the value went down (a discount) and plus when it went up (VAT, a pay rise, a markup). The percentage always refers to the original value, which is why you divide rather than add the percentage back.
Common error
The mistake almost everyone makes
A coat costs £210 in a 20% off sale. Adding 20% back gives £252 — and that is wrong.
The 20% was taken off the original price, not off £210. So £210 is 80% of the original: 210 ÷ 0.8 = £262.50. Adding the percentage back applies it to a smaller number, so you always land short. Check it forwards: £262.50 − 20% = £210
Multiplier lookup table
What to divide by, depending on which direction the change went. Increase and decrease are different numbers for the same percentage — 20% is 1.2 one way and 0.8 the other.
| Percentage | Divide by, to undo an increase | Divide by, to undo a decrease |
|---|---|---|
| 5% | 1.05 | 0.95 |
| 10% | 1.1 | 0.9 |
| 15% | 1.15 | 0.85 |
| 20% | 1.2 | 0.8 |
| 25% | 1.25 | 0.75 |
| 30% | 1.3 | 0.7 |
| 40% | 1.4 | 0.6 |
| 50% | 1.5 | 0.5 |
| 60% | 1.6 | 0.4 |
| 75% | 1.75 | 0.25 |
Three ways to work it out
All three give the same answer. Which one you want depends on whether you have a calculator, and whether you need to show your working. Teaching a class? The 18 exam questions and printable worksheet PDF are free and ungated.
Method 01
The multiplier method — fastest
Work out what fraction of the original you are left with, then divide by it. After a 15% discount you have 85% left, so the multiplier is 0.85. A £68 sale price was 68 ÷ 0.85 = £80. For an increase, add instead: a 15% markup gives 1.15, so £92 ÷ 1.15 = £80. This is the method the calculator above uses, and the one worth memorising — one division and you are done.
Method 02
The unitary method — what exams reward
Find what 1% is worth, then scale up to 100%. If £68 represents 85% of the original, then 1% is 68 ÷ 85 = £0.80, so 100% is 0.80 × 100 = £80. It takes an extra line, but it makes every step visible — which is exactly what mark schemes award method marks for. If you lose the thread in an exam, this is the one to fall back on, because a wrong final answer with correct working still scores.
Method 03
Estimate first, then check
Before you divide, decide roughly where the answer should land. Reversing a discount always makes the number bigger; reversing an increase always makes it smaller. If you started with £68 after a discount and got an answer under £68, you have divided the wrong way round. That single sanity check catches most real mistakes, and it is why the calculator prints a check line under every result.
When you need a reverse percentage
Removing VAT or sales tax. Almost every invoice and receipt shows a gross figure. Bookkeeping needs the net one, and the only way back is division — which is why "add 20% back" is the single most expensive habit in small-business accounting. On a £1,200 gross invoice it costs you £40 of VAT you have mis-stated.
Checking a sale price. Shops advertise the discount, not the original. Working backwards tells you whether "60% off" is a genuine reduction or a markup that was applied first.
Undoing a change in a spreadsheet. A column of post-change values needs =B2/(1-C2) for a decrease and =B2/(1+C2) for an increase. Subtracting the percentage in a formula produces the same error as doing it by hand, just faster and across every row.
GCSE reverse percentages. A named topic on the UK maths syllabus, and one that reliably appears in exams precisely because the intuitive answer is wrong. Getting the multiplier the right way round is most of the marks.
Worked examples
Sale price → original price
A jacket costs £80 in a 20% off sale. 100% − 20% = 80% remains, so the original price is 80 ÷ 0.8 = £100. You saved £20.
Gross price → net (VAT removed)
An invoice totals £120 including 20% UK VAT. Divide by 1.2: the net amount is £100 and the VAT portion is £20.
Salary after a rise → salary before
A salary is £42,000 after a 5% rise. 42,000 ÷ 1.05 = £40,000 before the rise.
Receipt total → pre-tax price
A US receipt shows $53.75 including 7.5% sales tax. 53.75 ÷ 1.075 = $50.00 before tax.
Part → whole
30 students passed, which was 40% of the class. 30 ÷ 0.40 = 75 students in total.
Exam mark → total marks
A score of 68% came to 51 marks. 51 ÷ 0.68 = 75 marks available on the paper.
The awkward cases, explained
A 100% decrease. The answer is not a very large number — there is no answer. A 100% discount leaves zero, and every possible original price also leaves zero, so the original is genuinely unrecoverable. Calculators that show ∞ here are reporting a division by zero as if it were maths.
A decrease of more than 100%. A price cannot fall by 130% and still exist. Almost always this means the change was an increase, so the calculator offers to switch rather than refusing outright.
A 0% change. The original equals the final value. The calculator says so in words and hides the "you saved" line, because saving nothing is not a saving.
Percentages above 100%. Perfectly valid in the third mode: 30 can be 150% of 20. The part is simply larger than the whole, which happens whenever something has grown past its baseline.
Rounding. Where a result does not land on an exact penny, the check line shows ≈ instead of =. That is deliberate honesty: £103.09 − 3% is £99.9973, not £100, and rounding the check to hide the difference would make the arithmetic look wrong to anyone who verified it by hand.
Frequently asked questions
How do you calculate a reverse percentage?
Divide the final value by the multiplier for the change. For a decrease, divide by (1 − p/100); for an increase, divide by (1 + p/100). For example, £80 after a 20% discount is 80 ÷ 0.8 = £100.
What is the reverse percentage formula?
Original value = Final value ÷ (1 ± percentage/100). Use minus for a decrease and plus for an increase.
Why do you divide by the multiplier instead of just subtracting the percentage?
Because the percentage was taken from the original value, not from the final one. Adding 20% back to £210 gives £252, but the true original is £210 ÷ 0.8 = £262.50. The bigger the percentage, the bigger the error.
Can I just add the percentage back on to get the original number?
No. Adding the same percentage back applies it to a smaller base, so you always land short of the original. Only division reverses the change exactly.
How do you find the original amount after a percentage increase?
Divide by 1 plus the percentage as a decimal. A salary of £120,000 after a 20% rise was £120,000 ÷ 1.2 = £100,000 before.
How do you calculate VAT backwards from a gross price?
Divide the gross price by 1 plus the VAT rate. At 20% UK VAT, a £120 gross price is £120 ÷ 1.2 = £100 net, so the VAT is £20.
How do you work out reverse percentages without a calculator?
Use the unitary method. If 40% is 30, then 1% is 30 ÷ 40 = 0.75, so 100% is 0.75 × 100 = 75. It is slower but it is what GCSE marks reward.
If 30 is 40% of a number, what is the number?
75. Divide the part by the percentage as a decimal: 30 ÷ 0.40 = 75.
What is the difference between a reverse percentage and a regular percentage?
A regular percentage calculation starts from the original and applies a change. A reverse percentage starts from the result and works back to the original, which is why it uses division rather than multiplication.
How do I find the original price if I know the price at 25% off?
Divide by 0.75. A £60 sale price at 25% off was £60 ÷ 0.75 = £80 originally.
What's the difference between percent and percentage points — and which does this calculator use?
This calculator works in percent, meaning relative change. A move from 10% to 15% is a rise of 5 percentage points but a 50% increase. The two are not interchangeable, and mixing them up is one of the most common errors in this topic.
Can the original value be recovered after a 100% decrease?
No. A 100% decrease leaves zero, and any original value would produce the same result, so the information is genuinely gone. The calculator explains this instead of showing an error.
All calculators
Percentage Guru is a growing family of percentage tools. The reverse calculator above is the flagship — these are the rest.
Prices & discounts
Spreadsheets & embed
Original price after 10% · 15% · 20% · 25% · 30% · 40% · 50% · 60% · 70% · 75% off.