Direct and Inverse Proportion

Two ways quantities can be linked

Many quantities in science and everyday life depend on each other. Sometimes when one goes up, the other goes up too. Other times, when one goes up, the other goes down. These two patterns are called direct proportion and inverse proportion, and being able to tell them apart — and write an equation for each — is a core skill for GCSE maths and beyond.

This lesson builds on basic ratio ideas. If sharing in a ratio or scaling recipes is still shaky, work through Ratios and Proportions first, then come back. We will also lean on the equation-handling from Solving Linear Equations.

Direct proportion

Two quantities are in direct proportion when they increase or decrease at the same rate. Double one and the other doubles; halve one and the other halves. We write this with the proportionality symbol:

$$ y \propto x \quad\Longrightarrow\quad y = kx $$

Here k is the constant of proportionality — a fixed number that links the two quantities. The graph of y = kx is always a straight line through the origin (0, 0), and its gradient is k.

Everyday example: the cost of petrol is directly proportional to the number of litres you buy. Buy twice as much fuel and you pay twice as much. If 1 litre costs £1.50, then k = 1.50 and the equation is cost = 1.50 × litres.

Finding k and using the equation

The whole method has three steps: write the equation, find k, then use it.

Worked example 1. y is directly proportional to x. When x = 3, y = 12. Find y when x = 7.

1. Write the relationship: y = kx.
2. Find k by substituting the known pair: 12 = k × 3, so k = 4.
3. Use the equation y = 4x. When x = 7, y = 4 × 7 = 28.

Notice that the ratio y ÷ x is constant in direct proportion: 12 ÷ 3 = 4 and 28 ÷ 7 = 4. That constant is k.

Inverse proportion

Two quantities are in inverse proportion when one increases as the other decreases, at a matching rate. Double one and the other halves. We write:

$$ y \propto \frac{1}{x} \quad\Longrightarrow\quad y = \frac{k}{x} $$

The graph of y = k/x is a smooth curve called a hyperbola — it drops steeply at first, then flattens out, and never quite touches either axis.

Everyday example: the time taken to finish a fixed job is inversely proportional to the number of workers. With twice as many workers, the job takes half as long (assuming everyone works at the same rate). If 1 worker takes 12 hours, then 2 workers take 6 hours, 3 take 4 hours, and so on — the product workers × hours stays at 12.

Finding k for inverse proportion

Worked example 2. y is inversely proportional to x. When x = 2, y = 6. Find y when x = 4.

1. Write the relationship: y = k/x.
2. Find k: 6 = k ÷ 2, so k = 12.
3. Use the equation y = 12/x. When x = 4, y = 12 ÷ 4 = 3.

Check the inverse pattern: x × y stays constant. 2 × 6 = 12 and 4 × 3 = 12. ✓

Telling them apart

The quickest test is to ask what happens to the product and the ratio of the two quantities:

Feature	Direct proportion	Inverse proportion
Equation	y = kx	y = k/x
When x doubles	y doubles	y halves
What stays constant	the ratio y ÷ x	the product x × y
Graph	straight line through origin	curve (hyperbola)
Example	cost vs litres of petrol	workers vs time for a job

If a table of values keeps the ratio fixed, it is direct. If it keeps the product fixed, it is inverse.

Worked example 3 — a real-world direct problem

A spring stretches in direct proportion to the weight hung from it (Hooke's Law). A 50 g weight stretches it 4 cm. How far will a 125 g weight stretch it?

• Equation: stretch = k × weight.
• Find k: 4 = k × 50, so k = 4 ÷ 50 = 0.08 cm per gram.
• Use it: stretch = 0.08 × 125 = 10 cm.

Worked example 4 — a real-world inverse problem

It takes 4 painters 9 days to paint a building. How long would 6 painters take, working at the same rate?

This is inverse: more painters, fewer days. The product painters × days is constant.

• Total work = 4 × 9 = 36 painter-days.
• With 6 painters: days = 36 ÷ 6 = 6 days.

Using the formula instead: y = k/x with k = 36 gives days = 36 ÷ 6 = 6. Same answer.

Reading values from a graph

Graphs make the difference between the two types obvious at a glance, and they are also a quick way to find missing values.

For a direct relationship y = 4x, the points (1, 4), (2, 8) and (3, 12) all sit on a straight line that passes through (0, 0). The gradient of that line equals k = 4, so you can read k straight off the slope: rise ÷ run = 4 ÷ 1 = 4. If a "direct proportion" graph does not go through the origin, it is not true direct proportion — that is a useful checking trick in exams.

For an inverse relationship y = 12/x, the points (1, 12), (2, 6), (3, 4) and (4, 3) lie on a curve that swoops down steeply and then levels off, never touching either axis. You can confirm it is inverse by multiplying each pair: 1 × 12 = 2 × 6 = 3 × 4 = 4 × 3 = 12. The constant product is k.

So three quick checks identify any proportion table without drawing anything: if y ÷ x is constant it is direct; if x × y is constant it is inverse; if neither is constant it is some other relationship entirely.

A caution

Not everything that grows together is proportional. A child's height and age both increase, but not at a constant rate, and the graph does not pass through the origin — so they are not in direct proportion. Always check that doubling one input really does double (or halve) the other before applying these equations.

Practice activity

Work these out, deciding first whether each is direct or inverse.

1. y ∝ x and y = 20 when x = 5. Find y when x = 8.
2. y ∝ 1/x and y = 9 when x = 4. Find y when x = 6.
3. 3 taps fill a tank in 40 minutes. How long would 5 taps take?
4. A car travelling at a steady speed covers 180 km in 3 hours. How far in 5 hours?
5. State which stays constant in each: (a) y ÷ x, (b) x × y — and match each to direct or inverse.

Answers: 1) k = 4, y = 32 2) k = 36, y = 6 3) total = 120 tap-minutes, 120 ÷ 5 = 24 minutes (inverse) 4) k = 60 km/h, 5 × 60 = 300 km (direct) 5) y ÷ x constant → direct; x × y constant → inverse.

Why this matters

Direct and inverse proportion describe how the world scales. Direct proportion governs prices, currency conversion, fuel use, and stretched springs; inverse proportion governs shared workloads, speed versus time for a fixed journey, gas pressure versus volume, and gear ratios. Recognising which type you are facing tells you instantly whether more means more or more means less — and the simple equations y = kx and y = k/x let you predict an exact answer. Engineers, scientists, and economists use these relationships every day, so mastering them now opens the door to a great deal of practical and scientific maths.