An equal percentage increase and decrease do not cancel each other. The second calculation uses a different starting value.
A 10% example
Start with 100.
100 increased by 10% = 110
110 decreased by 10% = 99The increase adds 10. The decrease removes 11 because 10% of 110 is 11. The final value is 99, a net decrease of 1% from the original 100.
Use multipliers to see the result
A 10% increase has a multiplier of 1.10. A 10% decrease has a multiplier of 0.90.
1.10 × 0.90 = 0.99The combined multiplier is 0.99, so the final value is 99% of the start.
For any equal rate r written as a decimal, the combined result is 1 − r². A 20% rise and fall gives 1 − 0.20² = 0.96, leaving a 4% net decrease.
The recovery percentage must be larger
If a value falls from 100 to 80, returning to 100 requires an increase of 20 on a baseline of 80:
20 ÷ 80 × 100 = 25%A 20% loss therefore needs a 25% gain to recover. A 50% loss needs a 100% gain.
This is arithmetic, not a forecast
The rule applies to account balances, prices, attendance, and other quantities. It does not predict investment returns or account for taxes, fees, deposits, or withdrawals.
For repeated positive changes, read how compound percentage increases work. To undo a known increase, use the reverse percentage formula.