Percentages trip people up because the same word covers three different questions. This page separates them: taking a percentage of a number, expressing one number as a percentage of another, and measuring the change between two numbers.
The three formulas
① Percentage of: A × X ÷ 100 (15% of 300 = 45). ② Ratio: A ÷ B × 100 (42 out of 200 is 21%). ③ Change: (New − Old) ÷ Old × 100 (8,000 → 9,500 is +18.75%). The denominator of a change is always the OLD value — that is why a rise from 100 to 150 is +50% but a fall from 150 back to 100 is −33.3%.
The two classic traps
Sequential percentages multiply, they do not add: up 10% then up 10% is 1.1 × 1.1 = 1.21, a 21% rise. And stacked store discounts work the same way: 30% off then 25% off pays 0.7 × 0.75 = 52.5% of the original — a 47.5% discount, not 55%.