Adding and removing the same percentage do not cancel because the second operation uses a different base. Reverse calculations must undo multiplication. First write the forward multiplier, then divide the final value by that multiplier. This makes both the formula and exceptional zero-factor case visible.
Reverse an increase by dividing by 1 + p/100
After a 25% increase, the multiplier is 1.25. If the final value is 100, the original is 100 ÷ 1.25 = 80. Subtracting 25% from 100 would give 75 because it uses 100, not 80, as the percentage base.
Reverse a decrease by dividing by 1 − p/100
After a 20% decrease, 80% of the original remains, so the multiplier is 0.8. A final value of 100 came from 100 ÷ 0.8 = 125. Adding 20% to 100 reaches only 120. The missing 5 reflects the changed base.
Check the reverse factor before dividing
A 100% decrease has multiplier zero: every starting value becomes zero, so a nonzero final value is impossible and zero cannot reveal a unique original. Rates over 100% create a negative multiplier in the simple arithmetic model. Treat domain-specific meanings separately.