The three percentage questions
Almost every percentage problem is one of three shapes, and confusing them is where the arithmetic goes wrong.
X% of Y is a share of a known total: 20% of 80 is 16. Multiply the total by the percentage over 100.
X is what % of Y runs the other way — you have two numbers and want the ratio between them: 16 out of 80 is 20%. Divide the part by the whole, then multiply by 100.
Percentage change compares a before and an after: going from 80 to 96 is a 20% increase. Subtract the original from the new value, divide by the original, multiply by 100.
Why a 20% rise then a 20% fall does not return you to the start
This trips up more calculations than any other percentage mistake. Take 100, add 20% and you have 120. Now take 20% off 120 and you get 96, not 100 — because the second percentage is measured against the larger number. Percentage changes are not symmetric, and they do not add up: two consecutive 10% rises are a 21% rise, not 20%.
The same applies to discounts. A 50% discount followed by a further 20% off is not 70% off — it is 60% off, because the second reduction applies to the already-reduced price.
Percentage points versus percent
If an interest rate moves from 4% to 5%, that is a rise of onepercentage point, but a 25% increase in the rate. Both descriptions are true and they mean very different things — which is why the distinction matters whenever a change to something already measured in percent is being reported.