Percentages and Percent Change: Three Common Errors
Percentages are the most everyday math — and the easiest to get wrong. Most mistakes are not arithmetic but using the wrong base.
Percent versus percentage points
Going from 10% to 12% is "up 2 percentage points" or "up 20%" (2 ÷ 10). Both are correct but mean different things; mixing them is a common source of misleading reporting.
Rule of thumb: percentage points are the difference between two percentages; percent is the relative change.
The base changes with direction
Up 50% then down 50% does not return to start: 100 → 150 → 75, a net 25% loss — because the rise is based on 100 and the fall on 150.
Likewise, after a 20% loss you need a 25% gain to break even (1 ÷ 0.8 − 1). That is the arithmetic behind "losses are easy, recovery is hard".
Discounts and markups are not symmetric
20% off then 10% off is 0.8 × 0.9 = 0.72 — 28% off, not 30%. Discounts multiply; they do not add.
Conversely, cost 60 and price 100 gives a 40% gross margin but a 66.7% markup. Same profit, different denominator — always ask which one is being quoted.
A quick sanity check
- Ask: "the denominator of this percentage is what?"
- When a change exceeds 100%, check whether it means "doubled" or "grew by one more times".
- Chain changes multiplicatively, never additively.
- When comparing two rates, use percentage points.
This article provides general information and calculation methods only. It is not medical, investment or legal advice; consult a qualified professional for your situation.