Direct and inverse proportion

The golden ratio of recipes

• A pancake recipe for 4 needs 200 g flour. How much for 10 people? $200 \div 4 = 50$ g per person, $50 \times 10 = 500$ g.
• That's direct proportion 正比例: double the people, double the flour. The ratio stays the same.

Direct proportion

• $y \propto x$ means $y = kx$ for some constant $k$ (the constant of proportionality 比例常数).
• If $y$ doubles, $x$ doubles too — they scale together.
• Find $k$ from a known pair, then use it.

Inverse proportion 反比例

• $y \propto \dfrac{1}{x}$ means $y = \dfrac{k}{x}$ — as one rises, the other falls.
• More workers → less time. More speed → less travel time.

Proportion to powers and roots

• Proportion can be to a square, cube, or square root:
• $y \propto x^2$: $y = kx^2$. If $x$ doubles, $y$ quadruples.
• $y \propto \sqrt{x}$: $y = k\sqrt{x}$.

Direct proportion is a straight line through the origin. Non-linear proportion (to a square, cube, or root) curves away.

Worked example

• $y \propto x^2$, and $y = 50$ when $x = 5$. Find $y$ when $x = 8$.
• $50 = k(5^2) = 25k \Rightarrow k = 2$.
• $y = 2(8^2) = 2(64) = 128$.

An inverse calculation

• Six equally productive workers take 10 days. With fixed work, $t=k/w$, so $k=tw=10(6)=60$ worker-days.
• Ten workers take $t=60/10=6$ days, assuming equal productivity. Do not use direct proportion: more workers should reduce the time.

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