Think multiplicatively: a \(20\%\) decrease is \(\times 0.8\), successive changes multiply, and reverse-percentage problems divide by the multiplier. Direct and inverse proportion are equations with a constant \(k\); similar shapes scale lengths by \(k\), areas by \(k^2\) and volumes by \(k^3\).
Contents: Ratio and percentages (M3.1-M3.8)Proportion, growth and similarity (M3.9-M3.11)Must-knowTrapsReal questions
Ratio and percentages (M3.1-M3.8)
- Share in the ratio \(2 : 3 : 7\): each part is total \(\div 12\).
- Changing a ratio: express amounts, add to one part, solve. Mixtures: track the pure quantity (e.g. grams of red paint).
- Percentage multipliers: \(+15\% \to \times1.15\); \(-25\% \to \times 0.75\). Original value \(=\) final \(\div\) multiplier.
- \(+x\%\) then \(-x\%\) is \(\times(1 - \frac{x^2}{10000})\): a net decrease. \(A\) is \(25\%\) more than \(B\) \(\Leftrightarrow\) \(B\) is \(20\%\) less than \(A\).
- Map scale \(1 : 25000\): \(1\) cm \(= 0.25\) km, so \(1\) cm\(^2 = 0.0625\) km\(^2\).
Proportion, growth and similarity (M3.9-M3.11)
- \(y \propto x^n\) means \(y = kx^n\); inverse proportion \(y = \frac{k}{x^n}\). Find \(k\) from the given pair, then substitute.
- Chains: \(p \propto \sqrt q\) and \(q \propto \frac1r\) give \(p \propto \frac{1}{\sqrt r}\).
- Compound growth: amount \(= P(1 + r)^n\); decay: \(P(1 - r)^n\). 'First below half' means find the least \(n\) with \(0.8^n < 0.5\) by trial (\(0.8^4 = 0.4096\)).
- Similar solids: areas in ratio \(4 : 9\) means lengths \(2 : 3\) and volumes \(8 : 27\).
Must-know (no formula booklet)
- \(\times(1 \pm \frac{p}{100})\)
- original \(=\) final \(\div\) multiplier
- length \(k\), area \(k^2\), volume \(k^3\)
- \(P(1+r)^n\)
Classic traps
- Two successive \(10\%\) rises are \(21\%\), not \(20\%\).
- 'Increase by \(50\%\) then decrease by \(50\%\)' gives \(75\%\) of the start.
- Compound interest is not simple interest: \(2000 \times 1.05^n\), not \(2000 + 100n\).
Worked example
Q. After a \(15\%\) rise a salary is \(27\,600\). Original?
A. \(27600 \div 1.15 = 24000\).