The three percentage problems — and how to recognize them
Every percentage question is one of three shapes. Recognizing which one you’re facing is the whole game:
| Question shape | Formula | Example |
|---|---|---|
| What is X% of Y? | Y × X/100 | 15% tip on 240 → 36 |
| X is what % of Y? | X/Y × 100 | 45 of 180 attendees → 25% |
| Change from A to B? | (B−A)/A × 100 | 80 → 92 → +15% |
The third one is where mistakes live: the denominator is always the starting value. A stock that falls 50% then rises 50% ends at 75% of where it began — because the two 50%s are measured against different bases.
Percentages in business: margin vs markup
The most expensive percentage confusion in small business: markup is profit measured against cost, margin is profit measured against price. A product bought at 100 and sold at 150 has a 50% markup but a 33% margin. Pricing “for a 40% margin” by adding 40% to cost actually yields a 28.6% margin — a silent profit leak. To price for a target margin, divide cost by (1 − margin): 100 ÷ 0.60 = 166.67 for a true 40% margin.
Stacked percentages don’t add
Sequential percentage changes multiply, never add. A 20% discount followed by an extra 10% off is not 30% off — it’s 1 − (0.80 × 0.90) = 28% off. Three years of 10% growth is not 30% but 33.1% (1.1³). This is also why “down 60% then up 60%” leaves you at 64% of the start. When a calculation chains percentages, convert each to a multiplier (±x% → 1 ± x/100), multiply them, and convert back at the end — the one habit that makes compound discounts, inflation series, and investment returns all come out right.