Working a percentage backward
Whenever a percentage is added to a value or taken off it, the percentage is calculated on the original, and the result is the original times a multiplier. Adding 20% multiplies by 1.2; taking 30% off multiplies by 0.7. Going back means dividing by that same multiplier.
The calculator needs three things: the final value you can see, the percent, and whether that percent was added (an increase, such as a tax or a raise) or taken off (a decrease, such as a discount). It returns the original value and the amount added or taken off, which is the gap between the two, plus a check line that applies the percent to the original again.
The diagram shows why the final value is the wrong base: the added part is 20% of the first segment, but only one sixth of the whole bar.
Reverse percentage formulas
Both divisions are exact; the result is rounded only for display. A percent of 100 or more taken off has no answer, because nothing can be recovered once the whole value is gone; the calculator shows a message instead.
Worked examples
- The percent was added, so the multiplier is 1 + 20 ÷ 100 = 1.2.
- Divide: 120 ÷ 1.2 = 100.
- Amount of the percent: 120 − 100 = 20, which is 20% of 100.
- A price of 299 on the same basis works out to 249.17 net, with 49.83 of VAT.
- Multiplier for a 5% increase: 1.05.
- Divide: 42,000 ÷ 1.05 = 40,000.
- The raise was 42,000 − 40,000 = 2,000.
- A 30% reduction leaves 70% of the price, so the multiplier is 0.7.
- Divide: 56 ÷ 0.7 = 80.
- The discount was 80 − 56 = 24, which is 30% of 80.
VAT, a consumption tax charged in many countries, is a common reason to run this calculation, but rates differ by country and by product. GOV.UK lists 20% as the UK standard rate, and the EU VAT Directive (Article 97) requires each member state’s standard rate to be at least 15%. Treat the examples here as arithmetic, not tax advice, and use the rate that applies to the sale in question.
Divisors for common percentages
| Percent | Added: divide by | Share of the final value that is the percent | Taken off: divide by |
|---|---|---|---|
| 5% | 1.05 | 4.76% | 0.95 |
| 10% | 1.1 | 9.09% | 0.9 |
| 15% | 1.15 | 13.04% | 0.85 |
| 20% | 1.2 | 16.67% | 0.8 |
| 25% | 1.25 | 20% | 0.75 |
| 30% | 1.3 | 23.08% | 0.7 |
| 50% | 1.5 | 33.33% | 0.5 |
The middle column answers “how much of a gross price is tax”: at a 20% rate it is one sixth of the final price, not a fifth.
The classic error and others
- Subtracting the percent of the final value. Taking 20% of 120 gives 24, and 120 − 24 = 96. Add 20% back to 96 and you get 115.20, not 120, which proves the answer is wrong. The 20% belongs to the original, so divide instead.
- Adding the percent back after a discount. A sale price of 56 plus 30% is 72.80, short of the true regular price of 80. A decrease is undone by dividing by 0.7, not by multiplying by 1.3.
- Picking the wrong direction. Treat the 30% off as if it had been added and the tool returns 43.08 instead of 80. Check which way the percentage went before you read the result.
- Reversing stacked discounts as one sum. A price of 72 after 20% off and then a further 10% off came from 100, because the two discounts amount to 28%. Dividing by 0.7 as if the total were 30% gives 102.86. Reverse each step in turn, or use the effective discount from the discount calculator.
The final value was usually rounded to the cent, so a reversed figure such as 249.1666… is the closest net price, not necessarily the exact one on the seller’s books. Round it to the cent and multiply back to check.
Questions people ask
How do I take VAT off a price quickly?
Divide the gross price by 1 plus the rate written as a decimal: by 1.2 for a 20% rate, by 1.1 for 10%. The tax itself is the gross price minus that result.
Why doesn’t the same percentage undo itself?
The base changes between the two steps. Raising 80 by 25% gives 100, but taking 25% off 100 gives 75. Undoing a 25% increase takes a 20% decrease, and undoing a 25% decrease takes an increase of 33.33%.
Does this work for things other than prices?
Yes, for any quantity that grew or shrank by a known percent. A town with 46,000 residents after 15% growth had 46,000 ÷ 1.15 = 40,000 before it.
Sources
- VAT rates, GOV.UK.
- VAT rates (VAT Directive), European Commission, Taxation and Customs Union.