Here's a puzzle that catches almost everyone: if a price goes up 20% and then down 20%, are you back to the original price? No — you end up lower. Understanding why is the key to getting percentage changes right.
The two formulas
To increase a value by a percent, multiply by one plus the percent as a decimal. To decrease, multiply by one minus it.
- Increase: value × (1 + percent ÷ 100). Increase 200 by 25% → 200 × 1.25 = 250.
- Decrease: value × (1 − percent ÷ 100). Decrease 200 by 25% → 200 × 0.75 = 150.
To find the percentage change between two numbers, use ((new − old) ÷ old) × 100.
The asymmetry trap
Take 100. Increase it by 20%: 100 × 1.2 = 120. Now decrease 120 by 20%: 120 × 0.8 = 96. You don't get back to 100 — you land at 96.
The reason is the base. The 20% increase was 20% of 100 (so +20). The 20% decrease was 20% of 120 (so −24). Different bases, different amounts. A percentage is always measured relative to the number it's applied to, and that number changed in between.
| Step | Calculation | Result |
|---|---|---|
| Start | — | 100 |
| +20% | 100 × 1.20 | 120 |
| −20% | 120 × 0.80 | 96 |
How to actually reverse a change
To undo a 20% increase, you don't subtract 20% — you divide by 1.20. 120 ÷ 1.20 = 100. In general, to reverse an increase of p%, divide by (1 + p ÷ 100); to reverse a decrease of p%, divide by (1 − p ÷ 100).
Where this bites in real life
- Sales: a price marked up 50% then put on a "50% off" sale is not back to cost — it's 25% below the marked-up price but still above cost only if the markup was on cost.
- Investing: a stock that drops 50% needs a 100% gain to recover, not another 50%.
- Salaries: a 10% pay cut followed by a 10% raise leaves you 1% below your original salary.
New to percentage change? Start with How to Calculate Percentages.