Why +20% Then −20% Doesn't Get You Back
Work one example to see why a 20% rise followed by a 20% fall leaves a lower number, and which percentage does take it back.
Math Tool
Percentage Calculator
Fill in any two of a starting value, a percentage change and an ending value to work out the third, and see the change that would undo it.
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Work Out the Missing Value
Fill in any two fields and leave the third empty. The empty one is worked out as you type.
Fill in any two of the three to work out the third.
How to Use the Percentage Calculator
- Fill in any two of the starting value, the percentage change and the ending value.
- Leave the third empty. It is worked out and shown in the result; the empty field stays empty.
- To work out a different one, clear that field. If all three are filled in, clear one.
- Optionally add a second change to see its effect after the first.
Reading the Result
The percentage is of the starting value. A 20% increase on 2,000 adds 20% of 2,000.
The change that undoes it is the percentage that takes the ending value back to the starting value. After a 20% increase it is a fall of 16.67%, not 20%, because it is taken of the larger value.
Working backwards divides the ending value by 1 plus the percentage. Subtracting the percentage from the ending value does not return the starting value.
A second change is taken of the value after the first, so a rise and an equal fall do not cancel.
Assumptions and What Is Not Included
- Each percentage is applied to its current base: the starting value, then the value after the first change.
- The values are plain numbers, in no currency or unit.
- Not included: tax, fees, and the rounding a shop or an invoice may apply, percentage points, growth over time (see the CAGR Calculator) and averages of percentages.
This is arithmetic on the numbers you enter, not advice. See the disclaimer.
Example
Take 1,00,000.00 raised by 20%: it becomes 1,20,000.00. A 20% fall of that is taken of 1,20,000.00, not of 1,00,000.00, so it gives 96,000.00, a net change of −4.00%. Adding the two percentages gives 0.00%, which is not the combined change. Working backwards: 2,400.00 after a 20% increase came from 2,000.00, and the change that undoes a 20% increase is −16.67%.
How It Is Calculated
Ending value = starting value × (100 + change) ÷ 100. Change = (ending ÷ starting − 1) × 100. Starting value = ending × 100 ÷ (100 + change).
The change that undoes a change p is −100 × p ÷ (100 + p). The calculation uses exact fractions and rounds only the figures shown, to two decimals.
Percentage Calculator FAQ
Why is the answer not filled into the empty field?
The empty field tells the calculator what to work out. If the answer were written into it, the page would no longer know which value you meant to find. To work out something else, clear that field.
Why does a 20% rise followed by a 20% fall not return to the start?
The 20% fall is taken of the larger value after the rise, so it removes more than the rise added. The change that does return to the start is a fall of 16.67%.
How do I find the value before a percentage increase?
Enter the ending value and the percentage change and leave the starting value empty. It is found by dividing by 1 plus the percentage, not by subtracting it.
Why does it say the answer is outside the range?
The two values you entered are valid, but the value worked out from them is outside what this calculator shows. Nothing is rounded or cut off to fit, so try values that are closer together.
Is this the percentage difference between two numbers?
No. The percentage change here is measured from the starting value. A percentage difference that treats both numbers alike is a different measure that this calculator does not provide.