Three questions, one operation
Almost every percentage problem is one of three, and confusing them is the source of nearly every mistake made with them.
The three formulas
Worked example: 15% of 200
- 200 × 15 ÷ 100 = 30
The other two on the same numbers: 30 is 15% of 200, and going from 200 to 230 is a 15% increase.
Increases and decreases do not cancel
This is the single most expensive misunderstanding in the list:
| Start | Change | Result |
|---|---|---|
| 100 | +50%, then −50% | 75 |
| 100 | −50%, then +50% | 75 |
| 100 | −20%, then +20% | 96 |
| 100 | −50%, then +100% | 100 |
The percentages apply to different bases, so they never reverse each other. A portfolio down 50% needs a 100% gain to recover, not another 50%.
Removing a percentage that is already included
To find the pre-tax figure from a total, divide rather than subtract. From $120 including 20% sales tax: 120 ÷ 1.20 = $100. Subtracting 20% gives $96, which is wrong by $4.
The same applies to a discounted price, a gross-to-net conversion, or any figure where the percentage was added to something smaller than the total you are holding.
Percentage points are not percentages
A rate moving from 4% to 5% has risen by one percentage point and by 25 percent. Both statements are true and they describe the same change.
Reports of interest rates, unemployment and market share use this distinction constantly, and mixing the two turns a modest move into a dramatic one or the reverse.
Mental shortcuts that hold up
X% of Y always equals Y% of X, which is occasionally very useful: 16% of 25 is awkward, while 25% of 16 is 4. For 15%, take 10% and add half of it. For 5%, halve the 10%.
Successive discounts
Two discounts do not add. 20% off then a further 10% off is not 30%: the second applies to the already-reduced price, giving 0.80 × 0.90 = 0.72, a total reduction of 28%.
The order does not matter — 10% then 20% gives the same 72 — but the difference against the intuitive 30% is real money on a large purchase. Shops rarely correct the assumption, and the sign that says both figures is not making a claim that either is additive.
Percentage of what, exactly
Most disputes about a percentage are really disputes about the base. A 3% commission on a $250,000 sale could be 3% of the price, of the profit, or of the amount above a threshold, and the three give $7,500, a much smaller figure, and something else again.
Before agreeing to any percentage, establish the denominator in writing. The rate is the part everyone discusses; the base is the part that decides the money.
Related calculators
- Percent off calculator — discounts specifically
- Percentage change calculator — the third formula on its own
- Basis point calculator — when a percent is too coarse a unit