The three formulas
X is what % of Y = X ÷ Y × 100
% change = (new − old) ÷ |old| × 100
For example, 15% of 200 is 15 ÷ 100 × 200 = 30. The number 30 is 25% of 120. And a move from 80 to 100 is a (100 − 80) ÷ 80 = 25% increase.
Why an increase and a decrease are not symmetrical
Going from 80 to 100 is a 25% rise, but going from 100 back to 80 is a 20% fall. The change is measured against the starting value, and the starting value is different in each direction. This is why a stock that falls 50% needs to gain 100% to get back to where it started.
Discounts and stacked percentages
Two discounts do not simply add up. Take 20% off and then a further 10% off and you pay 0.8 × 0.9 = 72% of the original price, a total discount of 28%, not 30%. To work out an original price from a sale price, divide by the fraction paid: a $60 item that is 25% off originally cost $60 ÷ 0.75 = $80.
Percent versus percentage points
If an interest rate rises from 4% to 5%, it has gone up by one percentage point, but it has risen by 25% in relative terms. News reports and financial documents mix the two, so it is worth checking which one is meant.
Everyday uses
- Working out sale prices and sales tax
- Comparing test scores or survey results
- Tracking growth in prices, salary, traffic or savings
- Checking commissions, markups and tips