Percentage change explained
Percentage change tells you how big a movement is compared with where you started. It is simple to calculate, but the way it behaves surprises many people.
Open the Percentage Change CalculatorThe formula
change % = (new − original) ÷ original × 100Positive means an increase and negative means a decrease. The denominator is always the original value, because the question is “how much has it moved relative to the start?”.
Example: a monthly bill goes from 80 to 100. The difference is 20, and 20 ÷ 80 = 0.25, so the bill rose 25%.
Why a rise and a fall are not symmetrical
If the bill then falls from 100 back to 80, the difference is again 20, but 20 ÷ 100 = 0.20. The fall is 20%, not 25%. The gap is the same, but the starting point changed.
The same effect explains why gains and losses of equal percentages do not cancel out. A 50% rise takes 100 to 150. A 50% fall from 150 takes you to 75, not 100. To recover from a 50% loss you need a 100% gain.
| Fall | Rise needed to recover |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 33.3% | 50% |
| 50% | 100% |
Applying a percentage to a number
To increase a number by a percentage, multiply by 1 plus the percentage as a decimal. To decrease it, multiply by 1 minus the decimal.
- Increase 200 by 15%: 200 × 1.15 = 230.
- Decrease 200 by 15%: 200 × 0.85 = 170.
This single-multiplier method also makes it easy to chain changes: multiply by each factor in turn rather than adding the percentages.
Percent versus percentage points
When a rate is itself a percentage, there are two ways to describe a change.
- A percentage-point change is the plain subtraction. A rate moving from 4% to 5% rose by 1 percentage point.
- A percent change is relative. That same move is 1 ÷ 4 = 25% higher.
Both statements are true, but they sound very different. When you read “rates rose 25%”, check whether the writer means the relative change. When you write about rates, say “percentage points” if you mean the subtraction.
When percentage change misleads
- Small starting values. Going from 2 to 4 is +100%, but it is only a difference of 2. Always look at the absolute numbers as well.
- A zero starting value. Change from zero is undefined, because you cannot divide by zero. Describe the difference in absolute terms instead.
- Negative values. A move from −10 to −5 is an improvement, but the usual formula gives a confusing sign. State the absolute change and be explicit about direction.
- Repeated changes. Growth of 10% a year for three years is 1.1 × 1.1 × 1.1 = 1.331, a 33.1% total rise, not 30%.