The three formulas
Nearly every percentage question is one of these:
| Question | Formula | Example |
|---|---|---|
| What is X% of Y? | Y × X ÷ 100 | 25% of 80 = 80 × 25 ÷ 100 = 20 |
| X is what % of Y? | X ÷ Y × 100 | 18 is what % of 72? 18 ÷ 72 × 100 = 25% |
| What is the % change from A to B? | (B − A) ÷ A × 100 | 36 to 45: 9 ÷ 36 × 100 = 25% increase |
The rest of this guide applies them to everyday situations.
Discounts and sale prices
To find the sale price, multiply by what you pay, not what you save.
- 20% off 60: you pay 80%, so 60 × 0.8 = 48.
- 35% off 140: you pay 65%, so 140 × 0.65 = 91.
Two discounts stack by multiplying, not adding. "30% off, then an extra 20% off" is 0.7 × 0.8 = 0.56, so you pay 56%: a 44% total discount, not 50%.
Tips and service charges
- 15% tip on 64: 10% is 6.40, 5% is 3.20, total 9.60.
- 20% tip: find 10% and double it. On 64, that is 12.80.
- Splitting the bill: add the tip first, then divide by the number of people.
Adding tax or a pay rise
To increase a number by a percentage, multiply by 1 plus the percentage as a decimal.
- Add 17% sales tax to 200: 200 × 1.17 = 234.
- A 4% pay rise on 3,250 a month: 3,250 × 1.04 = 3,380.
To decrease by a percentage, multiply by 1 minus it: a 15% cut to 3,000 is 3,000 × 0.85 = 2,550.
Percentage change: increase or decrease
Percentage change = (new − old) ÷ old × 100
- Rent goes from 1,200 to 1,290: (1,290 − 1,200) ÷ 1,200 × 100 = 7.5% increase.
- A share falls from 50 to 42: (42 − 50) ÷ 50 × 100 = −16%, a 16% decrease.
Always divide by the old (starting) value. Dividing by the new value is the most common mistake.
Reverse percentages: finding the original
When you know the number after a change, divide rather than multiply.
- Price including 20% tax is 119.99. Before tax: 119.99 ÷ 1.2 = 99.99.
- Sale price 48 after 20% off. Original: 48 ÷ 0.8 = 60.
- Salary after a 5% rise is 42,000. Before: 42,000 ÷ 1.05 = 40,000.
Subtracting the percentage from the final number gives the wrong answer. Taking 20% off 119.99 gives 95.99, not 99.99.
Why ups and downs do not cancel
Percentages are relative to the starting number, so:
- 100 up 50% = 150. Then down 50% = 75, not 100.
- An investment that falls 20% needs a 25% rise to recover: 80 × 1.25 = 100.
This matters for investments, prices and any number that changes more than once. It is also why compound interest grows faster than simple interest.
Percent vs percentage points
When an interest rate rises from 4% to 5%:
- It has risen by 1 percentage point.
- It has risen by 25 percent (1 ÷ 4 × 100).
News reports and adverts sometimes use whichever sounds more dramatic. When comparing rates, such as APR and APY, percentage points are usually clearer.
Mental maths shortcuts
- 10%: move the decimal point one place left. 10% of 370 = 37.
- 1%: move it two places. 1% of 370 = 3.7.
- 5%: half of 10%.
- 25%: divide by 4. 50%: halve it. 75%: take a quarter off.
- Percentages are reversible: 8% of 50 equals 50% of 8, which is 4. Swap them when one way is easier.
Using a calculator or spreadsheet
- Phone calculator: for "X% of Y", type Y × X ÷ 100. Many calculators have a % key, but they behave differently, so the formula is safer.
- Spreadsheet: percentage of:
=B2*C2with C2 formatted as a percentage. Percentage change:=(B2-A2)/A2, formatted as a percentage.
Key takeaways
- X% of Y = Y × X ÷ 100.
- Percentage change = (new − old) ÷ old × 100.
- Multiply by what you pay for discounts, and divide to reverse a percentage.
- Stacked percentages multiply, and a percentage point is not the same as a percent.