Direct & inverse proportion

#### Tier: #Higher

Description

Proportion describes a specific relationship between two variables.

Direct proportion: $y \propto x$ means $y = kx$ for a constant $k$.

• Graph: straight line through the origin
• Double $x$, double $y$

Inverse proportion: $y \propto \frac{1}{x}$ means $y = \frac{k}{x}$ for a constant $k$.

• Graph: reciprocal curve
• Double $x$, halve $y$

Other forms:

• $y \propto x^2 \implies y = kx^2$
• $y \propto \sqrt{x} \implies y = k\sqrt{x}$
• $y \propto \frac{1}{x^2} \implies y = \frac{k}{x^2}$

Finding $k$: substitute the given values to calculate $k$, then use the formula.

Example: $y$ is directly proportional to $x^2$. When $x = 3$, $y = 36$. Find $y$ when $x = 5$. $$y = kx^2 \implies 36 = k \times 9 \implies k = 4 \implies y = 4 \times 25 = 100$$

Common error: finding $k$ using the formula for the wrong type of proportion (e.g. using $y = kx$ when it should be $y = kx^2$).

Links

Substitution Changing the subject Proportion