Reverse Percentage Calculator
Use this reverse percentage calculator to work back from a final amount and a percent to the original, or from a part to the whole. It also covers 80 is 4 of what number, the percentage of what calculator question, removing VAT and six worked examples.
Example values are filled in. Replace them with your own.
Original value
80
Working
- 1 + 20 ÷ 100 = 1.2
- 96 ÷ 1.2 = 80
- Amount of change
- 16
A coat costs $68 after 15% off. What did it cost before the sale? Adding 15% back to $68 is the usual wrong turn. The sale price is 85% of the original, so the original is $68 ÷ 0.85 = $80.
Pick a mode, enter the final amount and the percent, and this reverse percentage calculator returns the original amount with the working shown. It reverses a percentage increase, reverses a percentage decrease, or finds the whole when you know that a number is a given percent of it, as in 80 is 4% of what number.
How reverse percentages work
Quick answer: divide the final amount by the multiplier, which is 1 plus the percent for an increase and 1 minus the percent for a decrease. A price of $96 after a 20% rise is $96 ÷ 1.20 = $80.
What are reverse percentages?
A reverse percentage question gives you the result of a percentage change and asks for the starting amount. It also covers the related question of finding the whole when you know a part and the percent it represents. Both run the usual calculation backwards, so they are sometimes searched as backwards percentage problems.
How to reverse a percentage increase or decrease
original = final / (1 + p / 100) (after an increase of p%)
original = final / (1 - p / 100) (after a decrease of p%)
- Write the percent as a decimal (15% is 0.15).
- Find the multiplier: 1 plus the decimal for an increase, 1 minus it for a decrease.
- Divide the final amount by the multiplier.
The percent was taken from the original, so the final amount equals the original times the multiplier.
How to find the original amount from a percentage
When a part is a known percent of an unknown whole, divide the part by the percent as a decimal. For 80 is 4% of what number, the whole is 80 ÷ 0.04 = 2,000. The same rule answers 16 is 40 percent of what number (16 ÷ 0.40 = 40) and 15% of what number is 3 (3 ÷ 0.15 = 20).
Reading questions like "80 is 4 of what number"
Searches such as 80 is 4 of what number leave out the percent sign, but they mean 80 is 4% of what number. The general pattern is x is y percent of what number, and the answer is x ÷ (y ÷ 100). Mode 3 of the calculator above does this in one step. You can check any answer by multiplying back: 2,000 × 0.04 = 80.
The non-calculator method
Treat the original as 100%. After a 20% increase, the final amount is 120%. If 120% is 96, then 1% is 96 ÷ 120 = 0.8, and 100% is 0.8 × 100 = 80. This unitary method gives the same answer as dividing by the multiplier, and it is easy to set out as written working.
Why you cannot subtract the percent from the final amount
Taking 20% off $96 gives $76.80, not $80. The 20% rise was worked out on the original $80, which is $16. Twenty percent of $96 is $19.20, a different amount, because it uses the larger base.
Removing VAT or sales tax from a total
Tax is a percentage increase, so removing it is a reverse percentage. A price of £120 including 20% VAT has a net price of £120 ÷ 1.20 = £100, so the VAT is £20, which is one sixth of the total. A US total of $108 including 8% sales tax comes to $108 ÷ 1.08 = $100 before tax.
Reversing two changes in a row
When a value went through two changes, divide by each multiplier, or by their product. A price of $108 after a 20% rise and then a 10% fall came from $108 ÷ (1.20 × 0.90) = $100. Do not undo them by taking 20% off and then adding 10%, which gives $95.04.
Check your answer by going forwards
When you reverse calculate a percentage, check it by applying the percentage forward to your answer. If the original is $80 and the sale was 15% off, then 80 × 0.85 = 68, which matches.
Where reverse percentages come up
To calculate the original value from a percentage, start from what you know: a price after a sale, a total including tax or a figure after growth. A rent of $1,595 after a 10% rise was $1,595 ÷ 1.10 = $1,450. What percent of what is a different question, because the percent is the unknown. The percent calculator on the home page solves that one.
Reverse percentages in Excel and Google Sheets
With the final amount in B2 and the percent in C2 as a decimal or percent-formatted cell, type =B2/(1+C2) to reverse an increase and =B2/(1-C2) to reverse a decrease. For a part and a percent, use =B2/C2.
Limits
A 100% decrease leaves nothing, and an increase of −100% or lower is not a real increase, so the original cannot be recovered. A percent of 0 gives no whole either. The calculator refuses all three and says why.
Reverse percentage examples
Multipliers to divide by, for common percentages:
| Percent | After an increase | After a decrease |
|---|---|---|
| 5% | 1.05 | 0.95 |
| 10% | 1.10 | 0.90 |
| 15% | 1.15 | 0.85 |
| 20% | 1.20 | 0.80 |
| 25% | 1.25 | 0.75 |
| 50% | 1.50 | 0.50 |
Example 1: sale price, $68 after 15% off
Given
Answer
Example 2: VAT removal, £120 including 20% VAT
Given
Answer
Example 3: salary, $52,500 after a 5% rise
Given
Answer
Example 4: population, 9,114 after 8.5% growth
Given
Answer
Example 5: 80 is 4% of what number
Given
Answer
Example 6: two changes, $108 after +20% then −10%
Given
Answer
Reverse percentage code
Both versions reproduce the examples above ($80, $50,000, 8,400, 2,000 and 100), reject non-numeric input and refuse the cases that cannot be reversed.
JavaScript
function finite(name, v) { if (typeof v !== "number" || !Number.isFinite(v)) throw new TypeError(`${name} must be a finite number`);}export function originalAfterIncrease(final, percent) { finite("final", final); finite("percent", percent); if (percent <= -100) throw new RangeError("percent must be greater than -100"); return final / (1 + percent / 100);}export function originalAfterDecrease(final, percent) { finite("final", final); finite("percent", percent); if (percent >= 100) throw new RangeError("percent must be less than 100: nothing can be recovered from a 100% decrease"); return final / (1 - percent / 100);}export function wholeFromPart(part, percent) { finite("part", part); finite("percent", percent); if (percent === 0) throw new RangeError("percent must not be 0"); return (part * 100) / percent;}Python
from decimal import Decimalimport mathdef _num(name, v): if isinstance(v, bool) or not isinstance(v, (int, float, Decimal)) or not math.isfinite(float(v)): raise ValueError(f"{name} must be a finite number") return Decimal(str(v))def original_after_increase(final, percent): final, percent = _num("final", final), _num("percent", percent) if percent <= -100: raise ValueError("percent must be greater than -100") return final / (1 + percent / 100)def original_after_decrease(final, percent): final, percent = _num("final", final), _num("percent", percent) if percent >= 100: raise ValueError("percent must be less than 100: nothing can be recovered from a 100% decrease") return final / (1 - percent / 100)def whole_from_part(part, percent): part, percent = _num("part", part), _num("percent", percent) if percent == 0: raise ValueError("percent must not be 0") return part * 100 / percentFor two changes in a row, divide by each multiplier in turn: originalAfterIncrease(originalAfterDecrease(108, 10), 20) returns 100. Divisions can produce long decimals, so round only for display. Use Decimal or integer cents for money.
Privacy and input safety
Values are calculated in your browser. The numbers you enter are not sent to a server or stored.
- Fields accept numbers only. Text, empty fields, NaN and Infinity are rejected with a message, and nothing you type is ever evaluated as code.
- A comma followed by exactly three digits is read as a thousands separator, so 1,200 is 1200. Any other comma is a decimal point, so 12,50 is 12.50, and a number starting with 0, such as 0,500, is read as a decimal. Mixed formats such as 1.234,5 and ambiguous entries such as 1,2,3 are refused instead of guessed. Spaces, including no-break spaces, between digit groups are ignored.
- Results are rounded only for display, so the working never compounds a rounding error.
Common reverse percentage mistakes
| Mistake | What goes wrong | What the calculator does |
|---|---|---|
| Subtracting the percent from the final amount | $96 less 20% is $76.80, not $80 | Divides by the multiplier |
| Adding the percent to a reduced price | $68 plus 15% is $78.20, not $80 | Divides by 0.85 |
| Mixing up increase and decrease | $68 treated as the result of a 15% rise gives $59.13, not $80 | Makes you choose the mode and shows the multiplier |
| Dividing by the percent alone | 96 ÷ 0.20 is 480, not 80 | Divides by 1 plus or minus the percent |
| Multiplying the part by the percent | 80 × 4% is 3.2; the whole is 80 ÷ 4% = 2,000 | Divides in the part-of-whole mode |
| Flipping signs to reverse two changes | $108 reversed as −20% then +10% gives $95.04, not $100 | Divides by the product of the multipliers |
| Reversing a 100% decrease | The final amount is zero, so there is nothing to recover | Refuses with a message |
| Rounding the multiplier | Using 1.08 for a 7.5% rise shifts the result | Uses the exact percent and rounds only the display |
Reverse percentage vs the other percentage questions
| Question | You know | You want | Formula |
|---|---|---|---|
| Reverse an increase | Final amount, percent increase | Original | final ÷ (1 + percent ÷ 100) |
| Reverse a decrease | Final amount, percent decrease | Original | final ÷ (1 − percent ÷ 100) |
| Whole from a part | Part, percent | Whole | part ÷ (percent ÷ 100) |
| Percent of a number | Whole, percent | Part | whole × percent ÷ 100 |
| What percent | Part, whole | Percent | part ÷ whole × 100 |
| Percentage change | Original, final | Percent change | (final − original) ÷ original × 100 |
The reverse rows are the only ones where the unknown is the starting amount. Everything else works forward from the starting amount.