How percent increase is calculated
A positive answer is an increase and a negative one is a decrease. The old value is always the one you divide by, because the change is measured against where you started.
For a decrease, use 1 − p ÷ 100 instead.
Worked example
Rent goes from $1,200 to $1,350. The change is 1,350 − 1,200 = 150, and 150 ÷ 1,200 = 0.125, so rent rose 12.5%. Going the other way, from $1,350 back to $1,200 is a drop of 150 ÷ 1,350 = 11.11%. The dollar change is the same, but the percent isn't, because the starting point is different.
The most common mistake: up and down don't cancel
A 50% rise followed by a 50% fall does not get you back to where you started. $100 up 50% is $150, and 50% off $150 is $75. You are 25% down. To undo an increase you need a smaller decrease, and to undo a decrease you need a bigger increase:
| Change | Multiply by | To get back, you need |
|---|---|---|
| +10% | 1.1 | −9.09% |
| +20% | 1.2 | −16.67% |
| +25% | 1.25 | −20% |
| +50% | 1.5 | −33.33% |
| +100% (doubling) | 2 | −50% |
| −10% | 0.9 | +11.11% |
| −20% | 0.8 | +25% |
| −25% | 0.75 | +33.33% |
| −50% (halving) | 0.5 | +100% |
The same thing trips people up when working backwards. If a price is $92 after a 15% rise, the original is 92 ÷ 1.15 = $80. Taking 15% off $92 gives $78.20, which is wrong.
Percent vs percentage points
When the values are themselves percentages, there are two ways to describe a change. A mortgage rate going from 3% to 4.5% rose by 1.5 percentage points (4.5 − 3), which is a 50% increase (1.5 ÷ 3). Both are correct, but they mean different things. News reports often mix them up. Choose "Percentages" above to see both.
Percent change vs percent difference
Percent change has a direction: it starts from the old value. Percent difference compares two values where neither is the starting point, such as two stores' prices, and divides by their average. 50 and 65 differ by 15 ÷ 57.5 = 26.09%. The percentage calculator has a percent difference box.
Limitations
- A change from zero can't be written as a percent, since you can't divide by 0. Give the actual change instead.
- When the old value is negative, like a loss turning into a profit, the result uses |old| and can be hard to read. The actual difference often says more.
- Changes over several years compound. A 30% rise over 3 years isn't 10% a year; the yearly rate is about 9.14%.