Percentage change = (new value − old value) ÷ old value × 100. A positive result is a percentage increase and a negative result is a percentage decrease. Always divide by the old (starting) value, which is the step people most often get wrong.
Going from 80 to 100 is (100 − 80) ÷ 80 = +25%. Going from 100 back to 80 is (80 − 100) ÷ 100 = −20%. The same 20-unit gap gives two different percentages because the starting value is different. Check your own numbers with the percentage calculator.
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The formula
Percentage change, also called relative change or percent change, compares a new value with an old one as a share of the old one:
Percentage change = (new − old) ÷ old × 100
- If the result is positive, it is a percentage increase.
- If the result is negative, it is a percentage decrease. Many people drop the minus sign and say "a 20% decrease".
- The old value is the base. If the old value is zero, the percentage change cannot be calculated.
This guide goes deep on percentage change. For the full set of percentage formulas, such as percent of a number, discounts and tips, see how to calculate percentages.
Percentage increase: worked examples
| Situation | Old | New | Calculation | Change |
|---|---|---|---|---|
| Monthly pay raise | $4,000 | $4,300 | (4,300 − 4,000) ÷ 4,000 | +7.5% |
| Rent goes up | $1,200 | $1,290 | (1,290 − 1,200) ÷ 1,200 | +7.5% |
| Website visitors | 800 | 1,000 | (1,000 − 800) ÷ 800 | +25% |
Steps: subtract the old value from the new value, divide the result by the old value, then multiply by 100.
Percentage decrease: worked examples
| Situation | Old | New | Calculation | Change |
|---|---|---|---|---|
| Sale price | $60 | $45 | (45 − 60) ÷ 60 | −25% |
| Share price falls | $200 | $140 | (140 − 200) ÷ 200 | −30% |
| Body weight | 90 kg | 81 kg | (81 − 90) ÷ 90 | −10% |
The result is negative, so it is a decrease. A drop from $60 to $45 is a 25% decrease.
The multiplier shortcut
To apply a percentage change directly, turn it into a multiplier:
- Increase by r%: new value = old × (1 + r ÷ 100). A 20% increase is × 1.20.
- Decrease by r%: new value = old × (1 − r ÷ 100). A 20% decrease is × 0.80.
Example: $250 increased by 12% is 250 × 1.12 = $280. $250 decreased by 12% is 250 × 0.88 = $220. Repeated changes multiply, so three years of 5% growth is 1.05 × 1.05 × 1.05 = 1.157625, an overall rise of about 15.76%, not 15%. This is the same idea as compound growth in how compound interest works.
Reverse percentages: find the original
If you know the final value after a percentage change, divide by the multiplier to get the original:
- A price is $180 after a 20% increase. Original = 180 ÷ 1.20 = $150 (not 180 − 20% of 180 = $144).
- A price is $80 after a 20% discount. Original = 80 ÷ 0.80 = $100 (not 80 + 20% of 80 = $96).
The percentage was taken from the original, so you must divide by the multiplier rather than apply the percentage to the new number.
Why up 50% then down 50% is not zero
Percentages are always measured against the current base. Start with 100:
- Increase by 50%: 100 × 1.5 = 150.
- Decrease by 50%: 150 × 0.5 = 75.
You end up 25% below where you started. A loss needs a bigger percentage gain to recover, because the gain is applied to a smaller base. The recovery needed after a loss L is L ÷ (1 − L):
| Loss | Left from 100 | Gain needed to get back |
|---|---|---|
| 10% | 90 | 11.11% |
| 20% | 80 | 25.00% |
| 25% | 75 | 33.33% |
| 33.3% | 66.7 | 50.00% |
| 50% | 50 | 100.00% |
| 75% | 25 | 300.00% |
This is why a 50% drop in an investment needs a 100% gain to recover, and why large drops are so hard to undo.
Percentage points vs percent
A percentage point is the plain difference between two percentages. Percent change is the relative difference. If an interest rate moves from 4% to 6%:
- It rose by 2 percentage points (6 − 4).
- It rose by 50% in relative terms ((6 − 4) ÷ 4).
Headlines often mix the two, so check which one a figure means before you compare.
Common mistakes
- Dividing by the new value instead of the old one.
- Assuming equal up and down percentages cancel. They do not.
- Adding successive percentages. Two 10% increases make 21%, not 20%, because 1.1 × 1.1 = 1.21.
- Reversing a percentage from the wrong number. Divide by the multiplier, do not subtract the percentage of the new price.
- Mixing percent and percentage points.
For salary comparisons, you can also work out take-home figures in the hourly wage to biweekly pay guide.
How to use the percentage calculator
Open the percent change tool
Choose the option for percentage increase or decrease.
Enter the old and new values
For example 80 and 100.
Read the result and the steps
A positive answer is an increase and a negative answer is a decrease. The formula is shown with every result.
Calculate your percentage change
Free, with the steps shown. No signup.
Open the Percentage CalculatorFrequently Asked Questions
Conclusion
Percentage change is the difference divided by the old value, times 100. Use multipliers to apply it, divide by the multiplier to undo it, and remember that up and down percentages do not cancel. Try your own numbers in the percentage calculator, or read how to calculate percentages for the other formulas.
This guide is for general information. Examples use rounded, illustrative numbers. For investment, tax or pay decisions, check the exact figures with the source or a qualified adviser.