How Reverse Percentages Work
A reverse percentage problem is one where you know the result after a percentage has been applied, and you need the original number. Typical questions include “40 is 25% of what number?”, “the sale price is 80 after 20% off; what was the original price?”, and “a value is 125 after a 25% increase; what was it before?”
The forward calculation is easy to remember: take a percentage of a number by converting the percentage to a decimal and multiplying. The reverse calculation divides instead of multiplying, and it uses a different denominator depending on whether you were given a part of the original, a discounted price, or a value after an increase. This article walks through each case and flags the situation that cannot be reversed at all. You can check the arithmetic in the reverse percentage calculator.
Reverse Percentage Calculator
Find the original number: when you know a part and its percentage, the price before a discount, or the value before an increase.
Result
The core idea
If a known amount is P% of some original number N, then:
Solving for N:
In words: convert the percentage to a decimal, then divide the known amount by that decimal. You are asking “this amount is P% of what?” rather than “what is P% of this amount?”
That is the right tool when the amount you have is a *part* of the original — a commission, a share, a portion of a total. Sale prices and “after an increase” values need a slightly different form, because 80 after 20% off is not 20% of the original; it is 80% of the original.
Case 1: A part is a known percentage of the original
Example: 40 is 25% of what number?
- Convert 25% to 0.25.
- Divide: 40 ÷ 0.25 = 160.
So 40 is 25% of 160. Check by going forward: 0.25 × 160 = 40.
The same method works for other percentages. If 18 is 12% of N, then N = 18 ÷ 0.12 = 150. If the percentage is greater than 100% — for example, 90 is 150% of N — then N = 90 ÷ 1.50 = 60. In that case the “part” is larger than the original, which is allowed.
If the percentage is 0, there is no unique original: 0 is 0% of every number. If you are using a calculator, that case should be reported as an error rather than as a made-up value.
Case 2: Finding the original after a decrease or a discount
After a discount of P%, the remaining fraction of the original is 1 − (P ÷ 100). The sale price is that fraction of the original, so:
Example: A sale price of 80 after 20% off
- The remaining fraction is 1 − 0.20 = 0.80.
- Original = 80 ÷ 0.80 = 100.
Check: 20% of 100 is 20, and 100 − 20 = 80.
A common mistake is to take 20% of the sale price and add it back (80 + 16 = 96). That answers a different question. Twenty percent was taken off the original, not off the reduced price, so you cannot recover the original by adding 20% of 80. This is the same trap as in How successive discounts stack, and it is why the denominator is 0.80, not an add-back on 80.
The discount calculator and the percentage off calculator start from the original price and compute the sale price. Reverse percentage starts from the sale price and works backwards.
Case 3: Finding the original after an increase
After an increase of P%, the new value is 1 + (P ÷ 100) times the original, so:
Example: 125 after a 25% increase
- The multiplier is 1 + 0.25 = 1.25.
- Original = 125 ÷ 1.25 = 100.
Check: 25% of 100 is 25, and 100 + 25 = 125.
The same pattern applies to taxes or markups if they are described as a percentage of the pre-increase amount. If 115 includes a 15% increase on the pre-tax (or pre-markup) figure, the original is 115 ÷ 1.15 = 100.
The case that cannot be reversed: a 100% reduction
If a value is reduced by 100%, the remaining fraction is 1 − 1 = 0, and the result is 0 no matter what the original was. You cannot divide by zero, and you cannot recover a unique original from a current value of 0 after a complete reduction.
A reduction of more than 100% would also fail in the same way (the remaining fraction would not be a usable positive scale factor for this recovery). If the current value is already 0 and the story is “this is what was left after 100% off,” the original is not determined by the arithmetic. A well-behaved calculator reports that as an error, which is the rule described in our calculation methodology.
A compact way to remember the three formulas
- Known part, known percentage of the original: divide the part by P/100.
- Known value after a decrease of P%: divide by (1 − P/100).
- Known value after an increase of P%: divide by (1 + P/100).
All three are “divide by the factor that was applied.” The factor is P/100 when the amount *is* that percentage; it is the remaining or increased fraction when the amount is what *remains after* the percentage.
Worked checks you can do by hand
- 15% of a number is 30. Original: 30 ÷ 0.15 = 200. Forward check: 0.15 × 200 = 30. This is the inverse of the examples in How to calculate a percentage.
- 20% off leaves 60. Original: 60 ÷ 0.80 = 75. Forward check: 0.20 × 75 = 15, and 75 − 15 = 60.
- 10% more than the original is 55. Original: 55 ÷ 1.10 = 50. Forward check: 50 + 5 = 55.
If the forward check does not return the number you started with, the factor in the denominator is the wrong one — most often because a discounted price was treated as if it were the discount amount itself.
When reverse percentages appear in practice
- Sale tickets that show only the discounted price and the percent off.
- A known share (“this fee is 12% of the bill”) when the bill itself is missing.
- A later measurement after a stated increase, when you want the baseline.
- Tax-inclusive totals, if the tax is a stated percentage of the pre-tax amount.
They are the wrong tool when you have two raw values and no percentage, or when you want a percentage change between two values. In those cases use the percentage change calculator or the basic “what percent” calculation instead.
Common mistakes
- Adding the discount back onto the sale price. 20% of 80 added to 80 is not the original after 20% off.
- Dividing by P when you should divide by 1 − P/100. If 80 is the price after 20% off, divide by 0.80, not by 0.20. Dividing by 0.20 would treat 80 as if it were the discount amount.
- Using the new value as if it were the original in a later step, which stacks errors in multi-step problems.
- Trying to reverse a 100% reduction. There is no unique original.
Frequently asked questions
How do you calculate a reverse percentage?
Divide the known amount by the factor that was applied. For a part that is P% of the original, that factor is P/100. For a value after a P% decrease, it is 1 − P/100. For a value after a P% increase, it is 1 + P/100.
A jacket is $80 after 20% off. What was the original price?
80 ÷ 0.80 = 100. The original price was $100.
Why can’t I just add 20% of $80 back?
Because the 20% was taken off the original price, not off $80. 20% of 80 is 16, and 80 + 16 = 96, which is too low.
What if the percentage is 100% off?
The remaining amount is 0 for every possible original, so the original cannot be recovered. The calculation is undefined.
Is this the same as percentage change?
No. Percentage change starts from an original and a new value and returns a percentage. Reverse percentage starts from a percentage and a later amount and returns the original value.