Reverse Percentage Calculator
- Original value
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- Amount added or removed
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- The common wrong answer
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Checked against: 120 after a 20% increase was 100; 80 after a 20% decrease was 100 Last reviewed:
Sources and method
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To reverse a percentage, divide the final number by the multiplier that was applied: after a 20% increase divide by 1.20, after a 20% decrease divide by 0.80. So 120 after a 20% rise was 100, not 96.
Reverse percentage questions come up whenever you know the result of a change but not the starting point: the price before a sale, a salary before a raise, a bill before tax was added, last year’s figure from this year’s growth. The trap is that the percentage was calculated on the number you are trying to find, so you cannot take it off the number you have.
The formula
Any percentage change multiplies the original by a fixed factor. A 20% increase multiplies by 1.20; a 20% decrease multiplies by 0.80. Undoing the change means dividing by the same factor:
After an increase: Original = Final ÷ (1 + p ÷ 100)
After a decrease: Original = Final ÷ (1 − p ÷ 100)
That is the whole method. Every reverse percentage problem is one division, once you have the right multiplier.
Worked examples
A price after a rise
A rail season ticket costs 2,875 after a 15% increase. Before the increase it was 2,875 ÷ 1.15 = 2,500. Check it forwards: 15% of 2,500 is 375, and 2,500 + 375 = 2,875.
A price after a discount
A jacket is 63 in a 30% off sale. The full price was 63 ÷ 0.70 = 90. The saving is 27 — which is 30% of 90, but 42.9% of 63. That asymmetry is why reverse percentages trip people up.
A salary after a raise
After a 4% pay rise you earn 46,800. Your old salary was 46,800 ÷ 1.04 = 45,000, so the raise was worth 1,800 a year.
Removing tax from a receipt
A receipt shows 240 including 20% VAT. The net price is 240 ÷ 1.20 = 200, and the VAT is 40. Note that 40 is one sixth of 240, not one fifth: at a 20% rate, the tax is 20/120 of the inclusive total.
Why subtracting the percentage gives the wrong answer
The instinct is to take the percentage off the final number: 120 minus 20% is 96. It feels symmetric, but it is not, because the two percentages are taken from different bases. The 20% increase was 20% of 100. Taking 20% of 120 removes 24, overshooting the original by 4.
The error grows quickly with the size of the change. At 5% it is barely visible; at 50% the shortcut says 50 when the real answer is 66.67. The table shows the gap for a final value of 100.
| Percentage | Undo an increase | Undo a decrease | True original (after increase) | Shortcut answer |
|---|---|---|---|---|
| 5% | ÷ 1.05 | ÷ 0.95 | 95.24 | 95.00 |
| 10% | ÷ 1.1 | ÷ 0.9 | 90.91 | 90.00 |
| 12.5% | ÷ 1.125 | ÷ 0.875 | 88.89 | 87.50 |
| 15% | ÷ 1.15 | ÷ 0.85 | 86.96 | 85.00 |
| 18% | ÷ 1.18 | ÷ 0.82 | 84.75 | 82.00 |
| 20% | ÷ 1.2 | ÷ 0.8 | 83.33 | 80.00 |
| 25% | ÷ 1.25 | ÷ 0.75 | 80.00 | 75.00 |
| 30% | ÷ 1.3 | ÷ 0.7 | 76.92 | 70.00 |
| 40% | ÷ 1.4 | ÷ 0.6 | 71.43 | 60.00 |
| 50% | ÷ 1.5 | ÷ 0.5 | 66.67 | 50.00 |
The same asymmetry explains why a 50% loss needs a 100% gain to recover, and why a 25% discount followed by a 25% price rise leaves you 6.25% below where you started.
Taking tax out of an inclusive price
Sales tax, VAT and GST are all percentage increases applied to the net price, so removing them is a reverse percentage problem. Divide the inclusive price by 1 plus the rate. The table splits an inclusive price of 100 at several common rates.
| Tax | Rate | Net price | Tax portion | Tax as a fraction of the total |
|---|---|---|---|---|
| UK VAT standard | 20% | 83.33 | 16.67 | 1/6 |
| UK VAT reduced | 5% | 95.24 | 4.76 | 1/21 |
| India GST standard | 18% | 84.75 | 15.25 | 1/6.56 |
| India GST merit | 5% | 95.24 | 4.76 | 1/21 |
| Germany VAT | 19% | 84.03 | 15.97 | 1/6.26 |
| Australia GST | 10% | 90.91 | 9.09 | 1/11 |
For India specifically, the GST calculator does this with the CGST and SGST split. Rates change, so check the current rate with the tax authority for anything you are filing.
Undoing more than one change
When a figure has been through several percentage changes, undo them in reverse order, one division each. Equivalently, multiply all the factors together and divide once. A price that went up 10% and then came down 10% to 99 started at 99 ÷ (1.10 × 0.90) = 99 ÷ 0.99 = 100.
The same idea, run forwards, is how stacked discounts work: two discounts of 20% and 10% multiply to 0.80 × 0.90 = 0.72, a 28% discount in total. The discount calculator handles that case directly.
Common mistakes
- Subtracting the percentage from the final figure. Always divide by the multiplier instead.
- Using the wrong direction. A sale price was reduced, so divide by (1 − p); a price including tax was increased, so divide by (1 + p).
- Treating percentage points as percentages. An interest rate going from 4% to 5% is a one percentage point rise but a 25% increase in the rate.
- Rounding too early. Keep the full multiplier (1.125, not 1.13) until the final step, or the answer drifts by a few pence.
For forward percentage problems — X% of a number, or the percentage change between two values — use the percentage calculator.
Frequently asked questions
How do I find the original price before a percentage increase?
Divide the new price by 1 plus the percentage as a decimal. A price of 120 after a 20% rise was 120 ÷ 1.20 = 100. Subtracting 20% of 120 instead gives 96, which is wrong because the 20% was worked out on the original, not on 120.
How do I find the original price before a discount?
Divide the sale price by 1 minus the discount as a decimal. A coat selling for 63 after 30% off was 63 ÷ 0.70 = 90 before the sale.
Why can I not just subtract the percentage from the final number?
Because the percentage was taken from a different number. A 20% increase on 100 adds 20, but 20% of the result (120) is 24. Taking 24 off lands you on 96, not 100. The bigger the percentage, the bigger the error.
How do I remove VAT from a price?
Divide the VAT-inclusive price by 1 plus the VAT rate. At the UK standard rate of 20%, a price of 240 is 240 ÷ 1.20 = 200 before VAT, so the VAT is 40. A shortcut at 20% is to divide by 6 to get the VAT portion directly.
What is the reverse percentage formula?
Original = Final ÷ (1 + p/100) after an increase, or Original = Final ÷ (1 − p/100) after a decrease, where p is the percentage applied.
Does reverse percentage work after two changes?
Yes, but you undo them one at a time, last change first. A price that rose 10% then fell 10% to 99 was 99 ÷ 0.90 = 110 before the fall, and 110 ÷ 1.10 = 100 before the rise.