The three questions behind every percentage problem
Almost every percentage question is one of three shapes. Part of a whole: 18% of 250 is 0.18 × 250 = 45. Part as a share: 45 out of 250 is 45 ÷ 250 × 100 = 18%. Change between two values: from 250 to 295 is (295 − 250) ÷ 250 × 100 = 18%. Recognising which one you are facing solves most of the difficulty.
Reverse percentages catch people out
If a jacket costs $80 after a 20% discount, the original price is not $96. The $80 represents 80% of the original, so the original is 80 ÷ 0.8 = $100. Adding 20% back to a reduced price always undershoots, because the percentage is taken from a smaller base. The same trap appears with sales tax: to strip 20% VAT from a gross price, divide by 1.2 rather than subtracting 20%.
Percentage points are not percentages
If an interest rate moves from 4% to 5%, that is a rise of one percentage point but a 25% increase. Financial and political reporting mixes the two constantly, and the difference is large enough to change the meaning of a headline.
Successive percentages do not add
A 10% rise followed by a 10% fall does not return you to where you started: 100 becomes 110, then 99. Multiply the factors instead of adding the percentages — 1.10 × 0.90 = 0.99, a net loss of 1%. This is why a 50% investment loss needs a 100% gain to recover — a quirk that also shows up in compound growth.