Percentages, Ratios and Proportional Reasoning
Three related tools
Percentages, ratios and proportion are the same idea in different clothing: a multiplicative comparison to a base. For percentages, distinguish “find a percentage of a number”, “express one number as a percentage of another”, and “reverse a percentage change”. Reverse questions divide by the multiplier: if a price after a 15% rise is 92, the original was 92 ÷ 1.15 = 80, not 92 − 15%. For ratios, identify whether the ratio is part-to-part (boys:girls) or part-to-whole (boys:total); convert to fractions of the total before sharing. Direct proportion scales both quantities by the same factor; inverse proportion keeps the product constant.
Traps
Percentages of different bases are not comparable, and successive percentage changes do not add: +20% then −20% gives 0.96 of the original, a 4% fall. “Percentage points” measure an arithmetic difference between percentages; “per cent” measures a relative change. Keep the order of a ratio as given, and never add ratios directly. In compound settings, apply each multiplier in sequence.
Worked example
A jacket costs 80 before tax. It is marked up 25%, then discounted 10% in a sale. Selling price = 80 × 1.25 × 0.90 = 90. Profit = 10, which is 10 ÷ 80 = 12.5% of cost. Note that a 25% markup followed by a 10% discount is not a 15% markup.
Rule of thumb: write the multiplier for each change (1.25, 0.90) and multiply; never add the percentages.