How to Calculate Percentages by Hand (The Only 3 Formulas You Need)
Percentage problems feel confusing because they hide which of three questions is being asked. Identify the question, and the formula is mechanical.
| The question | Formula | Example |
|---|---|---|
| What is X% of Y? | Y × X ÷ 100 | 15% of 80 = 12 |
| X is what % of Y? | X ÷ Y × 100 | 42 of 60 = 70% |
| % change from X to Y | (Y − X) ÷ X × 100 | 50 → 65 = +30% |
1. What is X% of Y? (discounts, tips)
Y × X ÷ 100. A 15% tip on an $80 dinner: 80 × 15 ÷ 100 = $12. A 30% discount on a £60 jacket: 60 × 30 ÷ 100 = £18 off, so £42.
2. X is what percent of Y? (scores, progress)
X ÷ Y × 100. Scored 42 out of 60: 42 ÷ 60 × 100 = 70%. Saved €3,400 toward a €5,000 goal: 68%.
3. Percentage change from X to Y (prices, growth)
(Y − X) ÷ X × 100. Rent going from $1,200 to $1,320: (120) ÷ 1200 × 100 = +10%. Note the trap: a 50% drop followed by a 50% rise does not return to the start — 100 → 50 → 75, because the second percentage applies to the smaller base.
Mental shortcuts worth keeping
10% is “move the decimal”: 10% of 84 is 8.4. Build others from it: 5% is half of that, 20% is double, 15% is 10% + 5%. And X% of Y always equals Y% of X — so 8% of 50 is the easy 50% of 8 = 4.
Questions people ask
How do I add a percentage to a number?
Multiply by (1 + rate): adding 8% tax to $50 is 50 × 1.08 = $54. Subtracting 8% is 50 × 0.92.
Why don’t percentage changes cancel out?
Because the second change applies to a new base: −50% then +50% gives 75% of the start, not 100%.