Percentage calculator

Nearly every percentage question is one of three

Percentages feel fiddly because the same word covers several different questions. In practice there are three, and once you can name which one you are facing, the arithmetic follows immediately.

• What is X% of Y? Multiply: 15% of 240 is 240 × 0.15 = 36. Used for tips, discounts, tax and commission.
• X is what percent of Y? Divide and scale: 36 out of 240 is 36 ÷ 240 × 100 = 15%. Used for scores, shares and hit rates.
• What is the percentage change from X to Y? Take the difference, divide by the original: from 240 to 276 is 36 ÷ 240 × 100 = a 15% increase. Used for prices, salaries and growth.

The two traps worth knowing

The first is that percentage changes do not cancel. A price rising 20% and then falling 20% does not return to where it started: 100 becomes 120, then 96. The second fall is taken from a larger base. To reverse a 20% rise you need a 16.67% fall.

The second is confusing percent with percentage points. If an interest rate moves from 4% to 5%, that is a rise of one percentage point but a 25% relative increase. Both statements are true and they describe very different things. News coverage of rates, taxes and polling gets this wrong constantly.

Doing it in your head

• 10% is a decimal point moved one place left. 10% of 340 is 34.
• 5% is half of 10%. 1% is a decimal point moved two places.
• Build the awkward ones: 15% is 10% + 5%; 17.5% is 10% + 5% + 2.5%; 40% is 4 × 10%.
• Percentages are reversible: 8% of 25 is the same as 25% of 8, which is 2. Flip whichever way is easier.
• 20% off means paying 80% — one multiplication instead of a subtraction.

Where each version shows up

Shopping and dining lean on 'X% of Y'. Exam marks, conversion rates and market share are 'X is what percent of Y'. Salaries, rents, energy prices and investment returns are almost always percentage change. If you are unsure which one you need, ask yourself what the 100% refers to — the answer names the calculation.