The four formulas
- X% of Y = X ÷ 100 × Y. 15% of 200 = 30.
- X is what % of Y = X ÷ Y × 100. 30 is 25% of 120.
- Percentage change from X to Y = (Y − X) ÷ X × 100. From 80 to 100 is +25%.
- Add or take off P%: Y × (1 + P ÷ 100) or Y × (1 − P ÷ 100). 250 + 10% = 275; 250 − 10% = 225.
A common trap: up then down is not back to the start
If a price goes up 20% and then down 20%, it does not return to where it began. £100 + 20% = £120; £120 − 20% = £96. Each percentage is taken of a different starting number.
Worked example – a sale price: a £85 jacket has 30% off. 30% of £85 = £25.50, so you pay £85 − £25.50 = £59.50 (or £85 × 0.7).
Worked example – a pay rise: your pay goes from £27,000 to £28,350. The change is £1,350, and £1,350 ÷ £27,000 × 100 = 5%.
Percentage points
When a percentage itself changes – for example an interest rate going from 4% to 5% – that is a rise of 1 percentage point, but a 25% increase in the rate. Both are correct; they just measure different things.
Where percentages show up
VAT, discounts, interest rates, pay rises, tips and exam marks all use percentages. For VAT, use the VAT calculator: removing VAT is a division, not a subtraction of 20%.
Reverse percentages
Sometimes you know the final amount and the percentage, and want the original. If a price after a 25% discount is £60, the original was not £60 + 25%. Instead divide by what is left: £60 ÷ 0.75 = £80. The same idea is used to take VAT out of a price.
Percent of a percent
Percentages multiply rather than add. Two discounts of 10% in a row are 19% off in total (0.9 × 0.9 = 0.81), not 20%. A rise of 10% a year for two years is 21%, because the second rise is on the higher amount.
Percentages in everyday money
Tips are often 10–15% of the bill; a 20% deposit on a £250,000 home is £50,000; a 3% pay rise on £30,000 is £900 a year. Each of these is the first sum above – X% of Y – so you can check them in a second.
Tip: type any numbers into the four boxes above – the answers change as you type, so you can try several values quickly.