Reverse Percentage Calculator
Find the original value before a percentage increase or decrease was applied. Enter the final value, the percentage change, and whether it was an increase or a decrease — the calculator reverses the change to reveal the starting number.
Find the Original Value from Final and Percentage
Reverse Percentage Formula
Where:
- \(\text{Final}\) — the value you have after the percentage change was applied
- \(p\) — the percentage change (e.g. 20 for 20%)
- \(\text{Original}\) — the starting value before the change
Dividing the final value by the growth or shrink factor converts the result back to the base it was calculated from. Note that you divide by \(1 \pm p/100\), not by simply subtracting or adding \(p\%\) of the final value — that is the most common mistake with reverse percentages.
Reverse Percentage – Calculation Example
Example: a price rose by 20% and is now 120. What was the original price?
\[ 1 + \frac{20}{100} = 1.20 \]
\[ \text{Original} = \frac{120}{1.20} = 100 \]
Result: the original value was 100.
Example 2: a price fell by 20% and is now 80. Then \(1 - 20/100 = 0.80\), so \(\text{Original} = 80 / 0.80 = 100\). A 20% decrease followed by a 20% increase does not return 100 — the decrease costs 20 units, but the increase only adds back 16.
How to Use the Reverse Percentage Calculator
- Enter the Final value — the number you have after the change (e.g. a discounted price or a tax-inclusive amount).
- Enter the Percentage change that was applied, as a plain number (e.g. 20 for 20%, not 0.2).
- Choose the Change type: Increase if the value went up, Decrease if it went down.
- Read the Original value — the number before the percentage was applied.
What Is a Reverse Percentage?
A reverse percentage (also called "finding the original value" or "working backwards from a percentage") answers the question: I know the number after a percentage change — what was it before? This comes up constantly: restoring a sale price to its full price, extracting the net price from a gross price that includes VAT or sales tax, or finding a company's revenue before a reported growth rate.
The key insight is that a percentage change is always calculated from the original value, not from the final one. When a price increases by 20%, the final price is 120% of the original — a multiplier of 1.20. To undo the change, you divide by that multiplier. Multiplying by 1.20 and dividing by 1.20 are inverse operations, so you land exactly back on the starting number.
This is also why you cannot simply subtract 20% of the final value. Taking 20% off 120 gives 96, not 100 — because 20% of 120 (24) is larger than 20% of 100 (20). The base matters, and reverse percentage is the method that respects the correct base.
If you need to go the other direction — from an original value to a new one — use the Percentage Change calculator, or the dedicated Percentage Increase and Percentage Decrease calculators. For everyday percentage problems in both directions, the general Percentage calculator is a good starting point.
Common Uses
- Sale prices: an item costs 80 after a 20% discount; the pre-sale price was 100.
- Tax-inclusive amounts: a gross price of 120 with 20% VAT means a net price of 100 (VAT is 20% of the net, not of the gross).
- Markups: a retail price of 138 with a 15% markup on cost means the cost was 120.
- Financial reports: revenue of 1.15 million after 15% growth means last year's revenue was 1 million.
Assumptions and Limitations
- The calculator assumes the percentage was applied exactly once to the original value, using the original as the base (the standard convention).
- A value cannot be decreased by 100% or more and remain positive — the calculator will warn you if you try a 100% decrease, since dividing by zero is undefined. Decreases over 100% would imply a negative original value, which rarely makes sense in practice.
- Compound changes must be undone one step at a time. If a value increased by 10% and then by another 10%, you cannot reverse it with a single 20% change: the combined multiplier is \(1.10 \times 1.10 = 1.21\), not 1.20.
- Results are rounded to 4 decimal places, which is enough for money and most practical applications; tiny rounding differences may appear with repeating decimals.
Quick Reference Table
| Change | Multiplier | To reverse |
|---|---|---|
| +10% | \(1.10\) | Divide by 1.10 |
| +20% | \(1.20\) | Divide by 1.20 |
| +50% | \(1.50\) | Divide by 1.50 |
| −10% | \(0.90\) | Divide by 0.90 |
| −20% | \(0.80\) | Divide by 0.80 |
| −25% | \(0.75\) | Divide by 0.75 |
Notice the asymmetry: reversing a 20% increase requires dividing by 1.20 (about a 16.67% reduction), while reversing a 20% decrease requires dividing by 0.80 (a 25% increase). Percentage changes are not symmetric in the two directions, and this table makes the pattern easy to check at a glance.
Frequently Asked Questions
How do I calculate a reverse percentage by hand without a calculator?
Convert the percentage to a decimal multiplier first, then divide. For an increase of p%, the multiplier is \(1 + p/100\); for a decrease, it's \(1 - p/100\). Divide the final value by that multiplier. Example: final value 250 after a 25% increase means \(250 / 1.25 = 200\). You can also set up a proportion: if 250 is 125%, then 1% is \(250 / 125 = 2\), and 100% is 200.
Why isn't reversing a percentage the same as adding or subtracting the same percentage?
Because the percentage was originally calculated from the starting value, not the final one. If a price rose 20% from 100 to 120, then 20% of 120 is 24, not 20 — subtracting it gives 96, not 100. To undo the change correctly, divide by the multiplier (1.20 for a 20% increase, 0.80 for a 20% decrease). This base mismatch is the single most common reverse percentage mistake.
How do I find the original price before VAT or sales tax was added?
Treat the tax rate as a percentage increase. If the gross price is 248 with 24% VAT, divide: \(248 / 1.24 = 200\). The net price is 200 and the tax is 48. Be careful with methods like the "price includes tax" shortcut of dividing by (100 + rate) then multiplying — that's exactly this division. Always confirm whether the quoted tax rate applies to the net amount (the usual convention) or the gross amount, since the base changes the result.
What is the difference between reversing a percentage increase and a percentage decrease?
The formulas differ only in the sign: after an increase of p%, divide by \(1 + p/100\); after a decrease, divide by \(1 - p/100\). The percentages themselves are not symmetric. Reversing a 20% increase means dividing by 1.20 (a 16.67% reduction from the final value), while reversing a 20% decrease means dividing by 0.80 (a 25% increase from the final value). Pick the wrong direction and the answer will be noticeably off.
Can a reverse percentage result in a negative or zero original value?
A decrease of exactly 100% makes the multiplier zero, so dividing is undefined — the final value would have to be 0, and the original cannot be recovered. Decreases greater than 100% produce negative multipliers and, with a positive final value, a negative original — which almost never makes sense for prices or quantities. If you're entering a discount above 100%, double-check whether you actually meant a percentage of the final value or a different problem altogether.
How do I reverse multiple percentage changes applied one after another?
Undo them in reverse order, one step at a time — you cannot combine them into a single percentage. Two consecutive 10% increases multiply to \(1.10 \times 1.10 = 1.21\), not 1.20, so reversing requires dividing by 1.21. For example, if a value rose 10% and then 10% again to reach 121, the original was \(121 / 1.21 = 100\). Work backwards from the last change to the first for exact results.
How do I check whether my reverse percentage answer is correct?
Apply the percentage change forward to your answer and see if you get the final value. If the original was 100 with a 20% increase, check: \(100 \times 1.20 = 120\). If the numbers match, the reversal is correct. If not, the most likely errors are using the wrong direction (increase vs. decrease), entering 0.2 instead of 20 for the percentage, or adding/subtracting the percentage directly from the final value instead of dividing by the multiplier.