MISSION · Algebra
Direct and Inverse Proportion
One goes up and so does the other — or exactly the opposite
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
In direct proportion the ratio of two quantities stays fixed; in inverse proportion their product does.
Direct proportion
\[y = kx \qquad \text{or} \qquad \frac{y}{x} = k\]Double one, double the other. Triple one, triple the other. The constant \(k\) is the constant of proportionality, and it’s the ratio that never changes.
Five apples cost £2, so ten cost £4 and twenty cost £8. The cost per apple — 40p — is \(k\).
The graph is a straight line through the origin, and that last part is not optional. A taxi charging £3 plus £2 a mile gives a straight line too, but it isn’t proportional: at zero miles you pay £3, not £0, and doubling the distance doesn’t double the fare. Proportional means zero-to-zero.
Inverse proportion
\[y = \frac{k}{x} \qquad \text{or} \qquad xy = k\]Double one and the other halves. The product stays fixed.
Six people take 4 hours to paint a fence, so twelve take 2 hours and three take 8. The total work — 24 person-hours — is \(k\), and it’s what stays constant however you rearrange the crew.
The graph is a curve that drops steeply and then flattens, approaching both axes without ever touching them. It can’t touch: \(x = 0\) would need infinitely many workers, and \(y = 0\) would mean finishing in no time at all.
Telling them apart
Multiply pairs, then divide pairs, and see which gives a constant.
| \(x\) | 2 | 4 | 8 |
|---|---|---|---|
| \(y\) | 12 | 6 | 3 |
Products: 24, 24, 24. Constant, so it’s inverse: \(y = \frac{24}{x}\).
Had the ratios been constant instead, it would be direct. If neither is, the relationship is something else — and plenty of real ones are.
Other powers
The same language extends. \(y \propto x^2\) means \(y = kx^2\): the area of a circle varies as the square of its radius. \(y \propto \frac{1}{x^2}\) is inverse-square, which is how gravity, light intensity, and radio signals weaken with distance — twice as far away means a quarter the strength, not half.
The method for problems
These questions follow one shape. Write the relationship with \(k\) in it, use the given pair to find \(k\), then use the completed formula to answer the question. Finding \(k\) first is the step people skip, and without it there’s nothing to substitute into.
Common mix-ups
- Calling any straight line proportional. It must pass through the origin.
- Mixing up which stays constant. Direct keeps the ratio, inverse keeps the product.
- Adding instead of scaling. Proportional means multiplying by the same factor.
- Forgetting to find \(k\) before answering.
Check yourself
\(y\) is inversely proportional to \(x\), and \(y = 15\) when \(x = 4\). Find \(y\) when \(x = 10\).
\(k = xy = 60\), so \(y = \frac{60}{x}\), giving \(y = 6\) when \(x = 10\).
LEVEL 3 · REFLECT
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