Percentage changes multiply rather than add, because each one is measured from a different starting point. A 20% rise followed by a 20% fall leaves you at 1.20 × 0.80 = 0.96 of where you started: 4% down, not back to even. And after a 50% fall you need a 100% rise to recover.
This is one of the most common errors in everyday maths, and it shows up in shop prices, salaries, investment returns and statistics. The fix is simple once you see why it happens.
Why the changes do not cancel
Start with 100. A 20% rise adds 20% of 100, giving 120. A 20% fall then removes 20% of 120, which is 24, giving 96. The second change was applied to a bigger number, so it removed more than the first one added.
Turn each change into a multiplier and the arithmetic becomes clean:
- A rise of p% multiplies by (1 + p/100). +20% is × 1.20.
- A fall of p% multiplies by (1 − p/100). −20% is × 0.80.
- Several changes in a row: multiply all the multipliers together.
1.20 × 0.80 = 0.96. The net effect is × 0.96, which is a 4% fall. The order makes no difference: −20% then +20% is 0.80 × 1.20, also 0.96.
| Changes | 100 becomes | Net change |
|---|---|---|
| +5% then −5% | 99.75 | −0.25% |
| +10% then −10% | 99.00 | −1% |
| +20% then −20% | 96.00 | −4% |
| +25% then −25% | 93.75 | −6.25% |
| +30% then −30% | 91.00 | −9% |
| +50% then −50% | 75.00 | −25% |
Recovering from a fall
The same logic explains why losses are harder to recover from than they look. After a fall of p%, you have (1 − p/100) of the original, and getting back needs a rise of:
Rise needed = 1 ÷ (1 − p/100) − 1
| Fall | Rise needed to recover | 100 becomes |
|---|---|---|
| −5% | +5.3% | 95.00 |
| −10% | +11.1% | 90.00 |
| −20% | +25.0% | 80.00 |
| −25% | +33.3% | 75.00 |
| −33.33% | +50.0% | 66.67 |
| −40% | +66.7% | 60.00 |
| −50% | +100.0% | 50.00 |
| −60% | +150.0% | 40.00 |
| −75% | +300.0% | 25.00 |
| −90% | +900.0% | 10.00 |
Small falls are recovered by roughly the same percentage. Large ones are not: a 50% fall needs a 100% rise, and a 75% fall needs 300%. This is why investors and businesses care so much about avoiding large drawdowns — it is not pessimism, it is arithmetic.
Combining several changes
Prices over several years
A product’s price rises 5%, then 8%, then 3% over three years. The total rise is 1.05 × 1.08 × 1.03 = 1.168, so 16.8%, not 16%. Adding the percentages always understates a run of rises and overstates a run of falls.
Average growth rate
If something grew 16.8% over three years, the average yearly rate is not 16.8 ÷ 3 = 5.6%. It is the rate that, compounded three times, gives 1.168: the cube root of 1.168 minus 1, which is about 5.3%. This is what “compound annual growth rate” means. Dividing the total by the number of years overstates the average.
A pay cut and a pay restore
An employer cuts pay by 10% during a difficult year, then “restores” it with a 10% rise. A 40,000 salary becomes 36,000 and then 39,600 — still 400 short. A true restoration from 36,000 needs a rise of 11.1%. You can check figures like this with the reverse percentage calculator: enter the current pay and the cut to get the original.
Stacked discounts
Shops use the same arithmetic in the other direction. “40% off, plus an extra 25% off” is 0.60 × 0.75 = 0.45 of the price, so 55% off in total, not 65%. The discount calculator combines up to two discounts and adds tax.
When adding is close enough
For small changes, adding gives nearly the right answer. Two rises of 1% compound to 2.01%, so “about 2%” is fine. The error grows with the size of the changes: for two changes of a and b percent, the gap between adding and multiplying is a × b ÷ 100 percentage points. At 2% and 3% that is 0.06 points; at 30% and 40% it is 12 points. As a rule of thumb, adding is fine for changes under about 5% each, and misleading above about 10%.
Quick check
- A share falls 30% and then rises 30%. Net: 0.70 × 1.30 = 0.91, so 9% down.
- A town’s population grows 10% a year for two years. Net: 1.10 × 1.10 = 1.21, so 21% up.
- A price falls 25%. To get back: 1 ÷ 0.75 − 1 = 33.3% rise.
For the change between any two numbers, use the percentage calculator; it always measures from the starting value.