Percentage
What you must know
A percentage is a fraction with denominator 100. Convert freely: 37.5% = 3/8, 16⅔% = 1/6, 62.5% = 5/8, 12.5% = 1/8. Memorising these pairs saves minutes. The core formulas are: percentage change = (change ÷ original) × 100, and successive changes multiply rather than add: a 20% rise then a 10% fall gives 1.20 × 0.90 = 1.08, i.e. +8%.
Also master: x% of y = y% of x; if A is x% more than B, then B is (x/(100+x))×100 percent less than A; and population/depreciation problems use the compound-interest-style formula P(1 ± r/100)ⁿ.
Traps
- Base confusion: "increased by 25% then decreased by 25%" is not zero change; it is 1.25 × 0.75 = 0.9375, a 6.25% net fall.
- The "more than / less than" asymmetry: if A is 25% more than B, B is 20% less than A, not 25%.
- Percentage points versus percent: a rate moving from 20% to 25% rises 5 percentage points but 25 percent.
- In successive discount or tax problems, never add the percentages.
Rule of thumb and example
Always ask "percent of what?" before computing. Write the base next to the fraction.
Worked example: The price of rice rises by 25%. By what percent must a family cut consumption to keep spending unchanged? New price factor = 5/4, so consumption factor must be 4/5, a reduction of 1/5 = 20%. Shortcut: reduction = (r/(100+r))×100 = 25/125 × 100 = 20%.