Take $100. Lose 20 per cent and you have $80. Gain 20 per cent back and you have $96, not $100. The two percentages look like they should cancel, and they do not, because they are measured against different bases: the fall was 20 per cent of $100, the rise was 20 per cent of $80.
The asymmetry gets worse as losses grow
The gap between the fall and the rise needed to undo it widens sharply:
- A 10 per cent fall needs an 11.1 per cent rise.
- A 20 per cent fall needs a 25 per cent rise.
- A 50 per cent fall needs a 100 per cent rise.
- A 90 per cent fall needs a 900 per cent rise.
The formula behind that column is short: the recovery required is loss ÷ (1 − loss). At small losses the two figures are close enough that the difference rarely matters. Past about 30 per cent they separate quickly, which is why the last row looks so extreme.
Where this actually bites
Investment returns
A portfolio that falls 40 per cent needs a 66.7 per cent gain to break even, not a 40 per cent one. This is also why average annual returns can mislead: a year of +50 per cent followed by a year of −50 per cent averages to zero, but leaves you down 25 per cent.
Discounts stacked on discounts
Two successive 20 per cent discounts are not 40 per cent off. The second is applied to the already-reduced price, so the total is 36 per cent off. The discount calculator handles the stacked case directly.
Prices described in both directions
If a price rose 25 per cent, it has to fall 20 per cent to return to the old figure, not 25 per cent. Any time the same change is described from the other end, the percentage changes with it.
The habit that avoids it
When a percentage change matters, convert it to a multiplier and multiply the multipliers. A 20 per cent fall is ×0.8, a 20 per cent rise is ×1.2, and 0.8 × 1.2 = 0.96, which is the 4 per cent shortfall the arithmetic actually produces. Multipliers chain correctly in any order; percentages do not chain at all.
To check any single step, the percentage increase calculator and the percentage calculator will both show the working.