Direct and Inverse Proportion

What you'll learn

How to recognise direct proportion and inverse proportion in worded questions.
How to use the unitary method to scale quantities up or down.
How to use formulae such as $y = k x$ , $\frac{k}{x}$ , and $\frac{k}{x^2}$ .
How to match proportional relationships to their graph shapes.

The basic idea: proportion

Proportion is about how two quantities change together.

For example, if one notebook costs £2, then 3 notebooks cost £6. The number of notebooks and the total cost are linked by a constant price per notebook.

Definition

Proportion

Two quantities are in proportion when one quantity changes in a predictable multiplicative way as the other changes.

The multiplier connecting them stays consistent.

A very useful starting method is the unitary method.

Definition

Unitary method

The unitary method means finding the value for 1 unit first, then multiplying to find the value for the number you need.

Worked example: scaling with one machine

A packing machine fills 750 cartons in 3 hours. How long would it take the same machine to fill 1250 cartons?

Example

Using the unitary method

Find how many cartons the machine fills in 1 hour.
$750 \div 3 = 250$
So the machine fills 250 cartons per hour. Now find how many hours are needed for 1250 cartons.
$1250 \div 250 = 5$
The time needed is 5 hours.

Tip

Sanity check

If the same machine has to fill more cartons, the time should increase. If your answer is less than 3 hours here, something has gone wrong.

Direct proportion

In direct proportion, both quantities increase or decrease together by the same scale factor.

If one quantity doubles, the other doubles.
If one quantity is divided by 5, the other is divided by 5.

Definition

Direct proportion

A quantity $y$ is directly proportional to $x$ if

y = k x

where $k$ is the constant of proportionality.

The constant of proportionality is the fixed multiplier connecting the two variables.

Key Idea

Direct proportion rule

For direct proportion, the ratio $yx\frac{y}{x}$ stays constant.

Worked example: cost of fruit

6 oranges cost £1.20. Work out the cost of 9 oranges.

Example

Direct proportion with money

Find the cost of 1 orange.
$1.20 \div 6 = 0.20$
Multiply by 9 to find the cost of 9 oranges.
$0.20 \times 9 = 1.80$
The cost is £1.80.

Common Mistake

Adding instead of scaling

Do not think “9 is 3 more than 6, so just add a bit”. Proportion questions are about multiplying or dividing by a scale factor, not usually adding.

Worked example: weight and number of sheets

A pack of 400 sheets of card weighs 1.6 kg. Work out the weight of 75 sheets.

Example

Scaling down, then up

Find the weight of 1 sheet in kg.
$1.6 \div 400 = 0.004$
Multiply by 75.
$0.004 \times 75 = 0.3$
The weight of 75 sheets is 0.3 kg.

Direct proportion with formulae

Sometimes the formula is already given. Then you usually substitute the value you know.

For example, $c = 0.35 n$ could mean the cost, in pounds, of buying $n$ identical items.

Worked example: using a direct proportion formula

The mass of a metal rod, $m$ grams, is directly proportional to its length, $l$ cm.

The formula is

m = 24 l

Find the length of a rod with mass 180 grams.

Example

Rearranging a direct proportion formula

Substitute $m = 180$ into the formula.
$180 = 24 l$
Divide both sides by 24.
$\div 24$
Calculate the length.
$l = 7.5$
The length is 7.5 cm.

Tip

What does the multiplier mean?

In $m = 24 l$ , the number 24 means each 1 cm of rod has mass 24 grams. That helps you check whether your answer is sensible.

Combining direct proportion with other information

Some questions give a total cost involving different items. You may need to use direct proportion first to find the cost of one type of item, then subtract.

Worked example: mixed items

2 packets of rice and 5 tins of soup cost £6.10 altogether.
4 packets of rice cost £3.20.
Work out the cost of 1 tin of soup.

Example

Using one item price to find the other

Find the cost of 1 packet of rice.
$3.20 \div 4 = 0.80$
Find the cost of 2 packets of rice.
$0.80 \times 2 = 1.60$
Subtract this from the total cost to find the cost of 5 tins of soup.
$6.10 - 1.60 = 4.50$
Divide by 5 to find the cost of 1 tin of soup.
$4.50 \div 5 = 0.90$
One tin of soup costs £0.90.

Inverse proportion

In inverse proportion, one quantity increases while the other decreases.

A common example is workers and time for a fixed job. If you double the number of equally efficient workers, the job takes half as long.

Definition

Inverse proportion

A quantity $y$ is inversely proportional to $x$ if

\frac{k}{x}

where $k$ is the constant of proportionality.

Key Idea

Inverse proportion rule

For inverse proportion, the product $x y$ stays constant.

Worked example: painters and days

3 painters take 8 days to paint a house. How long would 2 painters take, working at the same rate?

Example

Using total painter-days

Find the total amount of work in painter-days.
$\times 8 = 24$
Divide by the new number of painters.
$24 \div 2 = 12$
It would take 2 painters 12 days.

Common Mistake

Forgetting the direction

With inverse proportion, fewer workers means more time. If 2 painters somehow take less than 8 days, the answer cannot be right.

Worked example: machines and production time

5 identical printers take 6 hours to print a batch of leaflets. How long would 4 identical printers take to print the same batch?

Example

Same job, fewer machines

Find the total machine-hours.
$\times 6 = 30$
Divide by 4 machines.
$30 \div 4 = 7.5$
The time needed is 7.5 hours.

Common Mistake

When inverse proportion applies

This method assumes the job is fixed and every worker or machine works at the same rate. If the question suggests breaks, different speeds, or changing job size, you need to think more carefully.

Inverse proportion formulae

If the formula is given, substitute the value you know.

For example, $\frac{600}{q}$ means $p$ is inversely proportional to $q$ .

Worked example: using an inverse formula

$x$ is inversely proportional to $y$ , and

\frac{840}{y}

Find $x$ when $y = 35$ .

Example

Substituting into an inverse formula

Substitute $y = 35$ .
$\frac{840}{35}$
Calculate.
$x = 24$
The value of $x$ is 24.

Inverse square proportion

Sometimes a quantity is inversely proportional to the square of another quantity.

Definition

Inverse square proportion

A quantity $F$ is inversely proportional to the square of $x$ if

\frac{k}{x^2}

where $k$ is constant.

This appears in contexts such as force, light intensity, and distance.

Worked example: force and distance

The force between two objects is given by

\frac{80}{d^2}

where $d$ is the distance between them in cm. Find the force when the objects are 4 cm apart.

Example

Inverse square formula

Substitute $d = 4$ into the formula.
$\frac{80}{4^2}$
Square the distance first.
$4^2 = 16$
Divide.
$\frac{80}{16} = 5$
The force is 5 units.

Tip

Doubling in inverse square

If distance doubles in an inverse square relationship, the value becomes one quarter as large, because the denominator is multiplied by 4.

Recognising proportional graphs

Graphs are a quick way to identify the type of proportion.

The four most common graph shapes are shown below.

Four labelled graph shapes for direct proportion, inverse proportion, direct square proportion, and inverse square proportion

Definition

Asymptote

An asymptote is a line that a curve gets closer and closer to but does not meet.

For inverse proportion graphs, the axes often act as asymptotes.

Key graph facts

$y$ directly proportional to $x$ : straight line through the origin.
$y$ inversely proportional to $x$ : hyperbola, with branches in the first and third quadrants when $k$ is positive.
$y$ directly proportional to $x^2$ : upward parabola through the origin when $k$ is positive.
$y$ inversely proportional to $x^2$ : two branches above the x-axis when $k$ is positive, symmetric about the y-axis.

Worked example: matching graphs to statements

Four graphs are described below.

Graph P is a straight line through the origin.
Graph Q is a U-shaped parabola with its lowest point at the origin.
Graph R is a hyperbola with branches in the first and third quadrants.
Graph S has two branches above the x-axis, one on each side of the y-axis.

Match each graph to the correct proportional relationship.

Example

Identifying graph shapes

Direct proportion, $\propto x$ , is a straight line through the origin, so this is Graph P.
Direct square proportion, $\propto x^2$ , is a parabola, so this is Graph Q.
Inverse proportion, $\propto \frac{1}{x}$ , is a hyperbola with branches in the first and third quadrants, so this is Graph R.
Inverse square proportion, $\propto \frac{1}{x^2}$ , has two branches above the x-axis, so this is Graph S.

Common Mistake

Not checking the origin

A straight line only shows direct proportion if it passes through the origin. A straight line like $y = 2 x + 3$ is linear, but it is not direct proportion.

Exam technique

In the exam

Decide whether the quantities move in the same direction or opposite directions.
For direct proportion, find the unit value or use $y = k x$ .
For inverse proportion, keep the product constant, such as workers times days.
Substitute carefully into formulae, especially when there is a square like $x^2$ .
Check whether your answer makes sense: more workers should mean less time, but more items should usually mean more cost.

Self review

Check yourself

If 7 identical items cost £4.20, how would you find the cost of 11 items?
If 6 machines take 10 hours for a fixed job, should 3 machines take more or less time?
Which graph shape would you expect for $\frac{12}{x^2}$ ?

Direct and Inverse Proportion

What you'll learn

How to recognise direct proportion and inverse proportion in worded questions.
How to use the unitary method to scale quantities up or down.
How to use formulae such as $y = k x$ , $\frac{k}{x}$ , and $\frac{k}{x^2}$ .
How to match proportional relationships to their graph shapes.

The basic idea: proportion

Proportion is about how two quantities change together.

For example, if one notebook costs £2, then 3 notebooks cost £6. The number of notebooks and the total cost are linked by a constant price per notebook.

Definition

Proportion

Two quantities are in proportion when one quantity changes in a predictable multiplicative way as the other changes.

The multiplier connecting them stays consistent.

A very useful starting method is the unitary method.

Definition

Unitary method

The unitary method means finding the value for 1 unit first, then multiplying to find the value for the number you need.