Direct and inverse proportion: formulas and examples

For simple problems, find the value of one, then scale up.

Direct: 5 pens cost £3.50. One pen costs $£3.50 \div 5 = £0.70$, so 8 pens cost $8 \times £0.70 = £5.60$.

Inverse: 6 workers take 10 days to build a wall. That's $6 \times 10 = 60$ worker-days of work. 4 workers take $60 \div 4 = 15$ days. More workers, fewer days.

$k$ is the constant of proportionality. Every question follows the same three steps: write the formula, find $k$, then use it.

1. Direct: $y \propto x$, and $y = 12$ when $x = 4$. Then $12 = 4k$, so $k = 3$ and $y = 3x$. When $x = 10$, $y = 30$.

2. Squared: $y \propto x^2$, and $y = 50$ when $x = 5$. Then $50 = 25k$, so $k = 2$ and $y = 2x^2$. When $x = 3$, $y = 18$.

3. Inverse: $y \propto \tfrac{1}{x}$, and $y = 8$ when $x = 3$. Then $8 = \tfrac{k}{3}$, so $k = 24$ and $y = \tfrac{24}{x}$. When $x = 6$, $y = 4$.

4. Inverse square: a force $F$ is inversely proportional to $d^2$, and $F = 20$ when $d = 2$. Then $20 = \tfrac{k}{4}$, so $k = 80$. When $d = 4$, $F = \tfrac{80}{16} = 5$.

Sometimes you can reason directly:

• $y \propto x$: if $x$ triples, $y$ triples.
• $y \propto x^2$: if $x$ doubles, $y$ is multiplied by $2^2 = 4$.
• $y \propto \tfrac{1}{x}$: if $x$ doubles, $y$ halves.
• $y \propto \tfrac{1}{x^2}$: if $x$ doubles, $y$ is divided by 4.

In Example 4, $d$ doubled from 2 to 4, so $F$ was divided by 4: $20 \div 4 = 5$. Same answer.

• Direct proportion ($y = kx$): a straight line through the origin. A straight line that doesn't pass through the origin is not direct proportion.
• $y \propto x^2$: a curve through the origin that gets steeper.
• Inverse proportion ($y = \tfrac{k}{x}$): a curve that gets closer to both axes without touching them.

Exams sometimes ask you to match each type to its graph.

Direct: cost and quantity, distance and time at a steady speed, a recipe and the number of servings. Inverse: speed and journey time for a fixed distance (see speed, distance and time), and the number of people sharing a bill and each person's share. For splitting amounts in a ratio, use our ratio calculator.

• Using direct proportion when the situation is inverse (check: should the answer go up or down?).
• Forgetting to square or square-root $x$ when finding $k$.
• Giving the formula without $k$'s value when the question asks for "a formula for $y$ in terms of $x$".

1. 3 kg of apples costs £4.20. How much do 7 kg cost?
2. $y \propto x$ and $y = 21$ when $x = 3$. Find $y$ when $x = 8$.
3. $p \propto \tfrac{1}{q}$ and $p = 6$ when $q = 5$. Find $p$ when $q = 10$.
4. $A \propto r^2$ and $A = 12$ when $r = 2$. Find $A$ when $r = 5$.

Answers: 1. £9.80. 2. $k = 7$, $y = 56$. 3. $k = 30$, $p = 3$. 4. $k = 3$, $A = 75$.

More practice in our proportion questions.