The four formulas
- X% of Y: multiply Y by X and divide by 100. 15% of 80 = 80 × 15 ÷ 100 = 12.
- X is what percent of Y: divide X by Y and multiply by 100. 30 out of 120 = 30 ÷ 120 × 100 = 25%.
- Percentage change from X to Y: subtract, divide by the starting value, multiply by 100. From 50 to 75 = (75 − 50) ÷ 50 × 100 = +50%.
- Y increased by X%: multiply Y by (1 + X ÷ 100). 100 increased by 20% = 120. For a decrease, use (1 − X ÷ 100).
Everyday uses
- Sales and discounts: 30% off a $60 jacket is $18 off, so you pay $42. Use the 'decreased by' row.
- Sales tax: an 8% tax on $45 is $3.60. Use 'X% of Y'.
- Raises: going from $62,000 to $66,000 is a 6.45% raise. Use 'percentage change'.
- Grades: 42 out of 50 is 84%. Use 'X is what percent of Y'.
- Prices and inflation: see how much a price really went up between two years.
A common trap
A percentage increase and the same percentage decrease don't cancel out. If a $100 stock rises 50% to $150 and then falls 50%, it ends at $75, not $100. That's because each percentage is taken from a different starting number. The 'percentage change' row always uses the first number as the base.
Reverse percentages: finding the original number
If you know the price after a discount or the total after tax, divide instead of subtracting. A jacket costs $54 after 10% off: the original price is $54 ÷ 0.9 = $60, not $54 + 10% = $59.40.
The same works for tax and VAT. $113 including 13% VAT was $113 ÷ 1.13 = $100 before tax, and the VAT is $13.
Percentages in your head
- 10%: move the decimal point one place left. 10% of 250 is 25.
- 5%: half of 10%. 5% of 250 is 12.5.
- 1%: move the decimal point two places. 1% of 250 is 2.5.
- 25%: divide by 4. 50%: divide by 2.
- Swap them round: 8% of 50 is the same as 50% of 8, which is 4.