2.5Higher tier1MA1 · Section 2: Equations, formulae and identities

Direct and inverse proportion

Key learnings

Turn the proportion statement into an equation with a constant k
Direct: $y = kx$ . Inverse: $y = \dfrac{k}{x}$ . With a square: $y = kx^2$ or $y = \dfrac{k}{x^2}$ . Find $k$ from the given pair first.
Inverse means divide, not multiply
If $y$ is inversely proportional to $x$ , then as $x$ grows $y$ shrinks — you divide by $x$ (or $x^2$ ).

Drawn from real examiner reports.

Mistaking an inverse-proportion problem for a direct one

Multiplying by $x^2$ instead of dividing by it — the single most common error on this topic.

Forgetting to square

When the relationship involves $d^2$ , students use $d$ instead of $d^2$ .

Leaving the answer as a proportionality statement

Writing $y \propto \tfrac{1}{x}$ and stopping — a formula needs an equals sign and a value for $k$ .

Always find k explicitly and write the full equation
Substitute the given pair, solve for $k$ , write $y = \ldots$ , then answer the question.

Statement	Equation
$y$ directly proportional to $x$	$y = kx$
$y$ directly proportional to $x^2$	$y = kx^2$
$y$ inversely proportional to $x$	$y = \dfrac{k}{x}$
$y$ inversely proportional to $x^2$	$y = \dfrac{k}{x^2}$

Foundation

First step in any proportion question?

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