How to Find the Original Price Before a Discount or Tax
You see a final price and a percentage, and you want to know what it was before. A jacket costs 60 after a 25% discount. A receipt total includes 20% tax. A salary is now 51,500 after a 3% raise. In each case the natural first move is to add or subtract the percentage from the number in front of you, and in each case that move gives the wrong answer.
These are called reverse percentage problems. The fix is a single division, and once you see why it works you can handle discounts, taxes and raises with the same method.
Why adding the percentage back fails
A 25% discount is taken from the original price, not from the sale price. So the 25% is a quarter of a number you do not know yet. If you work out 25% of 60 and add it back, you are taking a quarter of the wrong number.
Test it: 25% of 60 is 15, and 60 + 15 = 75. Now apply the discount to 75: 75 × 0.75 = 56.25. That is not 60, so 75 cannot be the original price.
The division method
Write the change as a multiplier. A discount of 25% leaves 100% − 25% = 75% of the price, so the multiplier is 0.75. The original price multiplied by 0.75 gives the sale price, so to go backwards you divide.
original = final ÷ multiplierDiscount: multiplier = 1 − rate. Increase: multiplier = 1 + rate.
Worked example: the coat costs 60 after 25% off. The multiplier is 0.75, so the original price was 60 ÷ 0.75 = 80. Check: 80 × 0.75 = 60.
The amount saved is the original minus the sale price, 80 − 60 = 20. Notice that 20 is 25% of 80, not 25% of 60, which is exactly why the first guess was wrong.
Removing tax or VAT from a total
An increase works the same way, but the multiplier is above 1. If a price includes tax at 20%, the total is the pre-tax price times 1.20. To recover the pre-tax price, divide the total by 1.20.
Worked example: a bill totals 96 including a 20% tax. The pre-tax amount is 96 ÷ 1.20 = 80, and the tax portion is 96 − 80 = 16. A common slip is to take 20% of 96, which is 19.20. That overstates the tax, because 20% applies to 80, not to 96.
Tax rates differ by country, region and product, so use the rate printed on your receipt or invoice. The method does not change. For checking totals, try the VAT calculator.
Undoing a raise, a markup or growth
Any quantity that grew by a known percentage can be reversed the same way. A salary of 51,500 after a 3% raise was 51,500 ÷ 1.03 = 50,000 before it. A population of 1,320 after growing 10% was 1,320 ÷ 1.10 = 1,200.
| Situation | Multiplier | Final value | Original value |
|---|---|---|---|
| 25% discount | 0.75 | 60 | 60 ÷ 0.75 = 80 |
| 20% tax added | 1.20 | 96 | 96 ÷ 1.20 = 80 |
| 3% raise | 1.03 | 51,500 | 51,500 ÷ 1.03 = 50,000 |
| 10% growth | 1.10 | 1,320 | 1,320 ÷ 1.10 = 1,200 |
| 40% off | 0.60 | 45 | 45 ÷ 0.60 = 75 |
Checking your answer and handy shortcuts
- Always multiply back. Take your answer, apply the original percentage, and confirm you land on the number you started with.
- Use fractions when they are friendly. After a 20% discount you pay 4/5 of the original, so multiply the sale price by 5/4. Paying 60 after 20% off means the original was 60 × 5 ÷ 4 = 75.
- Sense-check the direction. After a discount the original must be larger than what you paid; after an increase it must be smaller.
- Watch for stacked discounts. Two discounts of 20% and 10% do not combine to 30%. Reverse them one at a time, last discount first.
When you have a sale price and want the full price quickly, the percent off calculator shows the saving, and the percentage calculator handles the general case.
- Percent Off Calculator: Enter the percent off and the original price to see the final price and your saving. Switch the mode to find the percentage discount when you already know both prices.
- VAT Calculator: Pick whether you are adding VAT to a price or removing it, enter the amount and the VAT rate, and see the price and the tax portion at once.
- Percentage Calculator: Pick the question you need answered, enter two numbers and get the result instantly. The guide below shows the formula behind each mode.
This article is general information, not professional advice. See our Disclaimer and Editorial Policy.
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