Proportional & Inverse Proportional

Q: y is proportional to x, and y = 9 when x = 3. What is x when y = 21?

7 — The constant is a = 9/3 = 3. Put y = 21 into y = 3x, and x = 7.

Q: y is inversely proportional to x, and y = 4 when x = 3. What is y when x = 6?

2 — The constant is a = xy = 3 × 4 = 12, so y = 12/6 = 2.

Q: If y = 24/x, what value of x gives y = 3?

8 — 3 = 24/x, so x = 24/3 = 8.

Q: On a 1:50000 map, how many km does 3 cm stand for?

1.5 km — 3 × 50000 = 150000 cm = 1500 m = 1.5 km.

Q: A car travels 12 km per litre, so on x litres it covers y = 12x km. How far can it go on 25 litres?

300 km — y = 12 × 25 = 300 km.

Q: A job takes 6 people 8 days. How many days does it take 4 people? (Number of people and number of days are inversely proportional.)

12 days — a = 6 × 8 = 48. With 4 people it takes 48/4 = 12 days.

Q: The graph of y = ax passes through (2, −6). What is a?

−3 — −6 = a × 2, so a = −3.

Where this is used

"Double your speed and you cover double the distance in the same time" — that is proportion. Map scales, the distance charge on a taxi meter, and how much fuel you burn all work the same way. Turn it around: "double the number of workers and the job takes half as long" — that is inverse proportion, and it shows up in production planning, traffic jams, and the link between the pressure and volume of a gas. Between them, these two are the simplest way mathematics describes how one quantity depends on another.

Start gently

Proportion means "twice as much in, twice as much out". If one notebook costs ¥120, two cost ¥240. Drawn as a graph it is a straight line through the origin.

Key points

What proportion is

When y is proportional to x, you can write y = ax (with a ≠ 0). Multiply x by k and y is multiplied by k as well. The number a is called the constant of proportionality.

What inverse proportion is

When y is inversely proportional to x, you can write y = a/x (with a ≠ 0). The giveaway is that x times y always comes out to the same number, a.

Finding the constant of proportionality

Suppose y is proportional to x, and y = 12 when x = 4. Then a = y/x = 12/4 = 3, so y = 3x.

The constant for an inverse proportion

Suppose y is inversely proportional to x, and y = 8 when x = 2. Then a = xy = 2 × 8 = 16, so y = 16/x.

Putting it to work on map scales

On a 1:25000 map, a length of 4 cm on the paper stands for a real distance of y = 25000 × 4 = 100000 cm = 1 km.

See it drawn

y = 2x (proportional) and y = 6/x (inversely proportional)

The plotted points are y = 6/x (inverse proportion): the product xy is always 6.

Check yourself 10questions

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Q1 y is proportional to x, and y = 9 when x = 3. What is x when y = 21?

Q2 If y = −2x, what is y when x = −4?

Q3 y is inversely proportional to x, and y = 4 when x = 3. What is y when x = 6?

Q4 If y = 24/x, what value of x gives y = 3?

Q5 On a 1:50000 map, how many km does 3 cm stand for?

Q6 A car travels 12 km per litre, so on x litres it covers y = 12x km. How far can it go on 25 litres?

Q7 A job takes 6 people 8 days. How many days does it take 4 people? (Number of people and number of days are inversely proportional.)

Q8 The graph of y = ax passes through (2, −6). What is a?

Q9 y is inversely proportional to x, and y = −4 when x = 5. What is the constant a?

Q10 Which point does the graph of y = 5x NOT pass through?

Jobs that use this

Architect $80k

Logistics and supply chain engineer $85k

Data analyst $90k

Architect$80k

Works with proportion and inverse proportion every day: scaling drawings, working out how much material a build needs, and figures such as heat conductivity.

Maths used: ProportionScaleAreaEquations

Logistics and supply chain engineer$85k

Uses proportional and inverse-proportional models to get the best out of fuel economy, load capacity, and shipping costs.

Maths used: ProportionOptimizationStatisticsGraph theory

Data analyst$90k

Uses correlation analysis to spot proportional relationships between variables, then builds models that forecast how a business will do.

Maths used: ProportionStatisticsRegression analysisLinear algebra