Percent Increase and Decrease: Mind the Base
Up 50% then down 50% doesn't bring you back to the start. Understanding the base-rate trap prevents a lot of mistakes in investing and shopping.
Percentages are among the most commonly miscalculated things in daily life, and the cause is almost always the same: using the wrong base.
The basic formulas
- Change = (new − original) ÷ original × 100%
- New = original × (1 + increase%)
- Original = new ÷ (1 + increase%)
The second line is the crux: a percentage is always relative to the original value. Verify quickly with the Percentage Calculator.
The classic trap: gains and losses aren't symmetric
Say a stock goes from $100 to $150 — a 50% gain. It then falls from $150 back to $100. That drop is (100 − 150) ÷ 150 = −33.3%, not −50%. Same $50 move, but the base shifted from 100 to 150.
Another trap: percentage points vs percent
If a rate rises from 4% to 5%, you can say it rose “1 percentage point” or that it rose 25% — (5 − 4) ÷ 4. Both are correct but mean different things, and news coverage mixes them to exaggerate. Always check which one is meant.
Practical cases
- Discounts: 20% off means paying 80% of the original price.
- Tipping: a 15% tip on a $200 bill comes to 200 × 1.15 = $230.
- Tax-inclusive prices: at a 5% tax rate, pre-tax = tax-inclusive ÷ 1.05 — not multiplied by 0.95.
- Investment return: (sale price − purchase price + dividends) ÷ purchase price × 100%.
For more tax and discount scenarios, use the Discount Calculator and the Percentage Calculator.