How to Calculate Percentage Change (and Why +20% Then −20% Isn't Zero)
Percentage change is one of the most used, and most misused, calculations in everyday life. Prices, salaries, test scores, website traffic and investment returns are all reported as "up 12%" or "down 5%". This guide shows the formula, walks through examples you can check by hand, and covers the three traps that catch most people: percentage points, reversing a percentage, and stacking several changes.
The formula
Percentage change compares a new value with an old (starting) value:
Percentage change = (new − old) ÷ old × 100
A positive result is an increase and a negative result is a decrease. The key detail is that you always divide by the starting value, not the new one and not the average.
Worked examples
| Situation | Old | New | Calculation | Change |
|---|---|---|---|---|
| Rent goes up | 800 | 1,000 | (1,000 − 800) ÷ 800 | +25% |
| Rent comes back down | 1,000 | 800 | (800 − 1,000) ÷ 1,000 | −20% |
| Monthly visitors | 120 | 150 | (150 − 120) ÷ 120 | +25% |
| Weight loss | 92 kg | 87.4 kg | (87.4 − 92) ÷ 92 | −5% |
Notice the first two rows: the same 200 difference is +25% going up but −20% going down, because the starting value is different. That asymmetry is behind most percentage confusion. You can check any of these with the percentage calculator.
Percent vs percentage points
When the values you are comparing are already percentages, such as interest rates, tax rates or exam pass rates, there are two different ways to describe the change:
- Percentage points are the simple difference. A rate that moves from 4% to 5% has risen by 1 percentage point.
- Percent is the relative change. The same move from 4% to 5% is a rise of (5 − 4) ÷ 4 = 25%.
Both statements are true, but they sound very different. News reports and sales material sometimes pick whichever sounds bigger, so it is worth checking which one is meant.
Reversing a percentage: finding the original value
A jacket costs 64 after a 20% discount. What was the original price? The tempting answer is to add 20% back: 64 × 1.2 = 76.80. That is wrong, because the 20% was taken from the original price, not from 64.
The correct method is to divide by the multiplier that was applied:
- After a 20% decrease, the price is 80% of the original, so original = 64 ÷ 0.80 = 80.
- After a 25% increase, the value is 125% of the original, so original = new ÷ 1.25.
The same idea removes sales tax or VAT from a price that already includes it: divide by (1 + tax rate). The sales tax calculator and the discount calculator handle both directions.
Why successive changes don't add up
If a share price rises 10% one day and falls 10% the next, you might expect to be back where you started. You are not: 100 × 1.10 × 0.90 = 99, a net loss of 1%. Each change applies to a different base, so you have to multiply the factors, not add the percentages.
| Changes | Multipliers | Net effect |
|---|---|---|
| +10%, then −10% | 1.10 × 0.90 = 0.99 | −1% |
| +20%, then −20% | 1.20 × 0.80 = 0.96 | −4% |
| −50%, then +50% | 0.50 × 1.50 = 0.75 | −25% |
| +5% three years in a row | 1.05 × 1.05 × 1.05 ≈ 1.158 | ≈ +15.8% |
The last row is compound growth: three 5% rises give more than 15% because each rise builds on the previous one. The same logic powers the compound interest calculator. A useful consequence: after a 50% loss, you need a 100% gain just to get back to the start.
Percentage difference vs percentage change
Percentage change has a clear "before" and "after". When you compare two values with no natural starting point, such as the price of the same item in two shops, some people use percentage difference: the absolute difference divided by the average of the two values. For 40 and 50 that is 10 ÷ 45 ≈ 22.2%, while percentage change from 40 to 50 is 25% and from 50 to 40 is −20%. Say which method you used so readers can follow your numbers.
Common mistakes
- Dividing by the new value. Always divide by the starting value for percentage change.
- Mixing up points and percent when the values are themselves rates.
- Adding back a discount instead of dividing by the multiplier.
- Adding successive percentages instead of multiplying the factors.
- Changes from a tiny base. Going from 2 sales to 6 is a 200% increase, which sounds dramatic but may be noise. Quote the raw numbers alongside the percentage.
Quick reference
- Increase by p%: multiply by (1 + p/100).
- Decrease by p%: multiply by (1 − p/100).
- Undo an increase of p%: divide by (1 + p/100).
- Undo a decrease of p%: divide by (1 − p/100).
- Several changes in a row: multiply all the factors, then subtract 1.