How to calculate a percentage

What a percentage is

Per cent means "per hundred". 35% is 35 parts out of every 100, which is the same as the decimal 0.35 or the fraction 35/100 (7/20 once reduced). That is all a percentage is: a ratio written as if the whole were 100. Because everything is scaled to 100, percentages make it easy to compare things of different sizes — a 12% pay rise means the same thing on a small salary as on a large one, even though the amounts in dollars are very different.

To turn a percentage into a decimal, divide it by 100 (move the decimal point two places to the left): 7.5% = 0.075. To turn a decimal into a percentage, multiply by 100: 0.6 = 60%. Every formula below is just this conversion plus one multiplication or division.

1. What is P% of a number?

Multiply the number by the percentage and divide by 100. The formula is part = number × P ÷ 100. For 20% of 150: 150 × 20 ÷ 100 = 30. It works the same way with decimals and with percentages above 100: 12.5% of 200 is 25, and 150% of 40 is 60.

An equivalent way is to convert the percentage to a decimal first: 20% = 0.2, and 0.2 × 150 = 30. Which order you use does not matter; the result is the same because multiplication can be done in any order.

Mental-maths shortcuts

• 10%: move the decimal point one place left. 10% of 340 = 34.
• 5%: half of 10%. 5% of 340 = 17.
• 1%: move the decimal point two places left. 1% of 340 = 3.4; then 3% is three times that (10.2).
• 25%, 50%, 75%: a quarter, a half and three quarters. 75% of 64 = 48.
• Swap the numbers: 8% of 50 is the same as 50% of 8, which is 4. This trick turns many awkward questions into easy ones.

2. What percent is A of B?

Divide the part by the whole and multiply by 100: percent = A ÷ B × 100. If you scored 45 out of 60 on a test: 45 ÷ 60 × 100 = 75%. If 18 of the 24 people in a group voted yes, that is 75%.

Be careful about which number is the whole. "What percent of 40 is 50?" has 40 as the whole, so the answer is 125% — more than 100% because the part is bigger than the whole. "What percent of 50 is 40?" is 80%. Swapping the two numbers changes the answer completely.

3. A is P% of what number?

This is the reverse question: you know the part and the percentage, and you want the whole. The formula is whole = A × 100 ÷ P. If 30 is 20% of a number, the number is 30 × 100 ÷ 20 = 150. If a deposit of $450 is 15% of the price, the price is $3,000.00.

A common version of this question involves a price that already includes a percentage: "the price after a 20% rise is $120 — what was it before?". Here the $120 is 120% of the original, so the original is 120 × 100 ÷ 120 = $100.00. Subtracting 20% from $120 gives $96.00, which is wrong — see the mistakes section below.

Adding and subtracting a percentage

To increase a number by P%, multiply by (1 + P ÷ 100). A $200 bill plus a 15% tip: 200 × 1.15 = $230.00. To decrease by P%, multiply by (1 − P ÷ 100): $200 minus 15% = $170.00. To find the percentage change between two numbers, use the percentage increase calculator: from 80 to 100 is +25%, and from 100 to 80 is -20%.

Percent, decimal and fraction table

Five mistakes to avoid

1. Subtracting a percentage to undo an increase. After a 25% rise to 125, taking 25% off gives 93.75, not 100. Divide by 1.25 instead.
2. Adding percentages of different wholes. A 20% discount followed by 10% off the new price is 28% off, not 30%.
3. Mixing up percent and percentage points. An interest rate rising from 4% to 5% went up by 1 percentage point, which is a 25% increase.
4. Using the wrong whole. "What percent of 40 is 50" (125%) and "what percent of 50 is 40" (80%) are different questions.
5. Comparing changes from different starting points. Prices going from 10 to 15 rose 50%; falling back from 15 to 10 is a -33.33% drop. The two values differ by 40% if you use the percentage difference, which treats both sides the same.

Percentages in everyday life

Shops use them for discounts (see the percent off calculator), governments for taxes such as VAT, banks for interest, and teachers for grades. In each case the same three formulas apply. A $64 jacket at 30% off costs $44.80; a £100 item with 20% VAT costs £120.00; a test score of 27 out of 30 is 90%.

When in doubt, write the question as "part = whole × percent ÷ 100" and fill in the two numbers you know. Then solve for the third. That one equation covers every case in this guide; the three formulas above are just the three ways of rearranging it.

Checking your answer

A quick sanity check catches most slips. A percentage below 100% of a number must give a result smaller than the number; above 100% it must be bigger. If 30 is supposed to be 20% of something, that something has to be five times larger than 30 — and indeed it is 150. Rounding is the other thing to watch: work with full precision and round only the final answer, otherwise small errors pile up across several steps.