A reverse percentage works backwards from a final value to the original value that produced it. The most familiar case is a discount: you know the sale price and the percent off, and you want the price before the reduction. The formula is: original = final ÷ (1 − rate), where the rate is written as a decimal.
Why you cannot just add the discount back
A common mistake is to take the discount percentage of the sale price and add it. This is wrong, because the discount was calculated from the original price. A 30% discount means the sale price is 70% of the original, so you must divide by 0.70, not add 30% of the sale price.
Worked example: a sale price of $84 with 30% off
- The sale price is 100% − 30% = 70% of the original, i.e. a factor of 0.70.
- Divide the sale price by that factor: 84 ÷ 0.70 = 120.
- The original price was $120. Check: 30% of 120 = $36, and 120 − 36 = $84. Correct.
Worked example: a price after a tax or increase
If a price is $138 after a 15% increase, the new value is 115% of the old one. Divide: 138 ÷ 1.15 = $120. Reverse percentages are not only about discounts; any percentage change can be undone this way.
Reverse percentage cheat sheet
| Situation | Factor | Formula |
|---|---|---|
| 30% off | 0.70 | final ÷ 0.70 |
| 20% off | 0.80 | final ÷ 0.80 |
| 15% increase | 1.15 | final ÷ 1.15 |
| 12% decrease | 0.88 | final ÷ 0.88 |
Finding the discount amount too
Once you have the original price, the amount saved is original − final. In the first example, 120 − 84 = $36. That is often the number people actually want when comparing deals.
It also helps to sanity-check with a quick estimate. A 30% discount means the original is roughly one-and-a-half times the sale price, because dividing by 0.70 increases the number by about 43% (1 ÷ 0.70 ≈ 1.43). So a $84 sale price should reverse to about $84 × 1.43 ≈ $120, which matches the exact answer. Estimation catches errors before you commit to a number.
Reverse percentages also appear in statistics and finance: GDP figures reported "after inflation", prices shown including tax, or profits quoted after a commission. Whenever a published number is described as "already adjusted", it is a final value, and the original is one division away.
Use the reverse mode of our percentage increase calculator to undo a percentage change, or read how to calculate the percentage of a number for the forward direction.