How to calculate percentages without second-guessing yourself

Three questions, three formulas

Almost every percentage problem is one of three questions: what is X% of Y, X is what percent of Y, or how much has a value changed in percentage terms.

• X% of Y: divide X by 100 and multiply by Y.
• X is what % of Y: divide X by Y and multiply by 100.
• Change: subtract the old value from the new, divide by the old, multiply by 100.

Worked examples

A 15% tip on a bill of 1,840: 15 divided by 100 is 0.15, and 0.15 times 1,840 is 276.

Scoring 34 out of 40 on a test: 34 divided by 40 is 0.85, so 85%.

Rent rising from 18,000 to 19,800: the change is 1,800, divided by 18,000 is 0.1, so a 10% increase.

The reversal trap

Adding 20% and then removing 20% does not return you to where you started. Add 20% to 1,000 and you get 1,200. Take 20% off 1,200 and you land on 960, because the second percentage applied to a bigger base.

To reverse an increase properly, divide by 1 plus the rate. 1,200 divided by 1.2 gives 1,000 exactly.

Percentage points versus percent

If an interest rate moves from 4% to 5%, that is a rise of one percentage point and a 25% increase. Newspapers mix these up constantly. When the quantity itself is a percentage, always say which one you mean.