Direct and Inverse Proportion

What you'll learn

How to tell whether a situation is direct or inverse proportion.
How to use unit rates, like “cost for one item” or “work done by one machine”.
How to substitute into proportion formulae such as $y = 0.4 x$ and $F=36x2F=\frac{36}{x^2}$ .
How to recognise the graph shapes for direct, inverse, square, and inverse square proportion.

1. The key idea: scaling

Proportion questions are about how two quantities change together.

For example, if one machine works at a steady speed, then making more bottles takes more time. If the number of bottles doubles, the time doubles too.

Definition

Unit rate

A unit rate means the amount for one unit, such as the cost of one banana, the weight of one sheet of paper, or the number of bottles made in one hour.

Example

Finding a unit rate first

A machine makes 800 cartons in 4 hours. How long will it take to make 1000 cartons?

A bar model shows the machine making 200 cartons each hour, so 1000 cartons is five equal hourly blocks.

Find how many cartons the machine makes in 1 hour.
$800 \div 4 = 200$
Divide the new total by the hourly rate.
$1000 \div 200 = 5$
So the machine will take 5 hours.

2. Direct proportion

Direct proportion means both quantities increase or decrease by the same scale factor.

If 5 items cost £2, then 10 items cost £4. You doubled the number of items, so the cost doubled.

Direct proportion means both quantities are multiplied by the same scale factor.

Definition

Direct proportion

Two quantities are directly proportional if multiplying one by a scale factor multiplies the other by the same scale factor. In algebra, this is written as $y = k x$ , where $k$ is the constant of proportionality.

Key Idea

Direct proportion test

If one quantity doubles, the other quantity doubles. If one quantity is divided by 10, the other quantity is also divided by 10.

Example

Direct proportion with money

It costs £1.20 to buy 8 apples. Work out the cost of 11 apples.

A unit-rate diagram shows 8 apples costing £1.20, then one apple, then 11 apples.

Find the cost of 1 apple.
$1.20 \div 8 = 0.15$
Multiply by 11 apples.
$11 \times 0.15 = 1.65$
So 11 apples cost £1.65.

Direct proportion with a formula

Sometimes the formula is already given. You just substitute the value you know.

Example

Using a direct proportion formula

The mass of a wire is $m$ grams and its length is $l$ cm. The formula is $m = 25 l$ . Find the length when the mass is 175 grams.

Substitute $m = 175$ into the formula.
$175 = 25 l$
Divide both sides by 25.
$\div 25 = 7$
The length is 7 cm.

3. Mixed direct proportion costs

Some questions give a total cost made from two different items. The trick is to find the cost of the known item first, subtract it, then divide what is left.

Example

Finding the cost of one item in a mixture

4 packets of rice and 3 tins of soup cost £5.10.

A shopping bar model separates the total cost into the cost of rice and the cost of soup. 5 packets of rice cost £2.25.
Find the cost of one tin of soup.

Find the cost of 1 packet of rice.
$2.25 \div 5 = 0.45$
Find the cost of 4 packets of rice.
$\times 0.45 = 1.80$
Subtract this from the total to find the cost of 3 tins of soup.
$5.10 - 1.80 = 3.30$
Divide by 3 to find the cost of 1 tin of soup.
$3.30 \div 3 = 1.10$
One tin of soup costs £1.10.

Common Mistake

Forgetting to subtract

Do not divide the full total by the number of tins if the total includes another item. First remove the cost of the other item.

4. Inverse proportion

Inverse proportion means as one quantity increases, the other decreases.

Jobs with workers or machines often work like this. If there are more identical workers, the job takes less time.

Definition

Inverse proportion

Two quantities are inversely proportional if multiplying one quantity by a scale factor divides the other by the same scale factor. In algebra, this is written as $y=kxy=\frac{k}{x}$ .

For work problems, use worker-days, machine-hours, or similar units.

Example

Painters and days

It takes 3 painters 8 days to paint a house. How long would it take 4 painters to paint the same house?

A worker-days grid represents the same total job being shared by different numbers of painters.

Find the total amount of work in painter-days.
$\times 8 = 24$
Share the same work between 4 painters.
$24 \div 4 = 6$
It would take 4 painters 6 days.

Tip

Sanity check

For inverse proportion, more workers or machines should mean less time. If your answer goes up when you added more workers, check your method.

5. Inverse proportion formulae

If you are given a formula like $x=1000yx=\frac{1000}{y}$ , substitute the value carefully.

Example

Substituting into an inverse formula

$x$ is inversely proportional to $y$ and $x=900yx=\frac{900}{y}$ . Find $x$ when $y = 45$ .

Substitute $y = 45$ into the formula.
$x=90045x=\frac{900}{45}$
Divide.
$x = 20$
So the value of $x$ is 20.

6. Square and inverse square proportion

Sometimes the proportion involves $x^2$ , which means “ $x$ squared”.

Directly proportional to $x^2$ means $y=kx^2$ .
Inversely proportional to $x^2$ means $y=kx2y=\frac{k}{x^2}$ .

Example

Inverse square formula

The force between two objects is $F$ and the distance between them is $d$ cm. The formula is $F=48d2F=\frac{48}{d^2}$ . Find $F$ when $d = 4$ .

The inverse-square setup shows two objects separated by a distance of 4 cm before substituting into the formula.

Substitute $d = 4$ into the formula.
$F=4842F=\frac{48}{4^2}$
Square 4 first.
$F=4816F=\frac{48}{16}$
Divide.
$F = 3$

Common Mistake

Not squaring first

In $F=48d2F=\frac{48}{d^2}$ , you must square the distance before dividing. For $d = 4$ , use 16, not 4.

7. Recognising the graphs

The symbol $∝\propto$ means “is proportional to”.

Here are the key graph shapes:

The four key proportion graph shapes can be recognised from their overall form and position on the axes.

$\propto x$ : a straight line through the origin.
$\propto \frac{1}{x}$ : a curved graph with two branches, usually in the first and third quadrants when the constant is positive.
$\propto x^2$ : a U-shaped parabola through the origin.
$\propto \frac{1}{x^2}$ : two curved branches above the $x$ -axis when the constant is positive.

Example

Matching graph shapes

Four sketches are labelled P, Q, R, and S.

The labelled sketches P, Q, R and S match the four graph descriptions in the example. P is a straight line through the origin.
Q is a U-shaped curve through the origin.
R has two curved branches in opposite quadrants.
S has two curved branches above the $x$ -axis.

Match each sketch to the correct proportion statement.

$y$ is directly proportional to $x$ : choose P, because direct proportion gives a straight line through the origin.
$y$ is directly proportional to $x^2$ : choose Q, because squaring gives a U-shaped parabola.
$y$ is inversely proportional to $x$ : choose R, because the graph has two opposite branches.
$y$ is inversely proportional to $x^2$ : choose S, because $x^2$ is always positive, so the graph sits above the $x$ -axis.

Exam technique

In the exam

Decide first: direct means “both change the same way”; inverse means “one goes up, the other goes down”.
For word problems, find a unit rate or a total amount of work, and keep the units clear.
For formula questions, substitute carefully; if you see $x^2$ , square before you divide or multiply.

Self review

Check yourself

If 6 identical machines take 5 hours, should 3 machines take more time or less time?
What graph shape do you expect for $y = k x$ ?
In $F=80d2F=\frac{80}{d^2}$ , what should you do to $d$ before dividing 80?

Direct and Inverse Proportion

What you'll learn

How to tell whether a situation is direct or inverse proportion.
How to use unit rates, like “cost for one item” or “work done by one machine”.
How to substitute into proportion formulae such as $y = 0.4 x$ and $F=36x2F=\frac{36}{x^2}$ .
How to recognise the graph shapes for direct, inverse, square, and inverse square proportion.

1. The key idea: scaling

Proportion questions are about how two quantities change together.

For example, if one machine works at a steady speed, then making more bottles takes more time. If the number of bottles doubles, the time doubles too.

Definition

Unit rate

A unit rate means the amount for one unit, such as the cost of one banana, the weight of one sheet of paper, or the number of bottles made in one hour.

Example

Finding a unit rate first

A machine makes 800 cartons in 4 hours. How long will it take to make 1000 cartons?

A bar model shows the machine making 200 cartons each hour, so 1000 cartons is five equal hourly blocks.

Find how many cartons the machine makes in 1 hour.
$800 \div 4 = 200$
Divide the new total by the hourly rate.
$1000 \div 200 = 5$
So the machine will take 5 hours.

2. Direct proportion

Direct proportion means both quantities increase or decrease by the same scale factor.

If 5 items cost £2, then 10 items cost £4. You doubled the number of items, so the cost doubled.

Direct proportion means both quantities are multiplied by the same scale factor.

Definition

Direct proportion

Key Idea

Direct proportion test

If one quantity doubles, the other quantity doubles. If one quantity is divided by 10, the other quantity is also divided by 10.

Example

Direct proportion with money

It costs £1.20 to buy 8 apples. Work out the cost of 11 apples.

A unit-rate diagram shows 8 apples costing £1.20, then one apple, then 11 apples.

Find the cost of 1 apple.
$1.20 \div 8 = 0.15$
Multiply by 11 apples.
$11 \times 0.15 = 1.65$
So 11 apples cost £1.65.

Direct proportion with a formula

Sometimes the formula is already given. You just substitute the value you know.

Example

Using a direct proportion formula

The mass of a wire is $m$ grams and its length is $l$ cm. The formula is $m = 25 l$ . Find the length when the mass is 175 grams.

Substitute $m = 175$ into the formula.
$175 = 25 l$
Divide both sides by 25.
$\div 25 = 7$
The length is 7 cm.

3. Mixed direct proportion costs

Some questions give a total cost made from two different items. The trick is to find the cost of the known item first, subtract it, then divide what is left.

Example

Finding the cost of one item in a mixture

4 packets of rice and 3 tins of soup cost £5.10.

A shopping bar model separates the total cost into the cost of rice and the cost of soup. 5 packets of rice cost £2.25.
Find the cost of one tin of soup.

Find the cost of 1 packet of rice.
$2.25 \div 5 = 0.45$
Find the cost of 4 packets of rice.
$\times 0.45 = 1.80$
Subtract this from the total to find the cost of 3 tins of soup.
$5.10 - 1.80 = 3.30$
Divide by 3 to find the cost of 1 tin of soup.
$3.30 \div 3 = 1.10$
One tin of soup costs £1.10.

Common Mistake

Forgetting to subtract

Do not divide the full total by the number of tins if the total includes another item. First remove the cost of the other item.

4. Inverse proportion

Inverse proportion means as one quantity increases, the other decreases.

Jobs with workers or machines often work like this. If there are more identical workers, the job takes less time.

Definition

Inverse proportion

Two quantities are inversely proportional if multiplying one quantity by a scale factor divides the other by the same scale factor. In algebra, this is written as $y=kxy=\frac{k}{x}$ .

For work problems, use worker-days, machine-hours, or similar units.

Example

Painters and days

It takes 3 painters 8 days to paint a house. How long would it take 4 painters to paint the same house?

A worker-days grid represents the same total job being shared by different numbers of painters.

Find the total amount of work in painter-days.
$\times 8 = 24$
Share the same work between 4 painters.
$24 \div 4 = 6$
It would take 4 painters 6 days.

Tip

Sanity check

For inverse proportion, more workers or machines should mean less time. If your answer goes up when you added more workers, check your method.

5. Inverse proportion formulae

If you are given a formula like $x=1000yx=\frac{1000}{y}$ , substitute the value carefully.

Example

Substituting into an inverse formula

$x$ is inversely proportional to $y$ and $x=900yx=\frac{900}{y}$ . Find $x$ when $y = 45$ .

Substitute $y = 45$ into the formula.
$x=90045x=\frac{900}{45}$
Divide.
$x = 20$
So the value of $x$ is 20.

6. Square and inverse square proportion

Sometimes the proportion involves $x^2$ , which means “ $x$ squared”.

Directly proportional to $x^2$ means $y=kx^2$ .
Inversely proportional to $x^2$ means $y=kx2y=\frac{k}{x^2}$ .

Example

Inverse square formula

The force between two objects is $F$ and the distance between them is $d$ cm. The formula is $F=48d2F=\frac{48}{d^2}$ . Find $F$ when $d = 4$ .

The inverse-square setup shows two objects separated by a distance of 4 cm before substituting into the formula.

Substitute $d = 4$ into the formula.
$F=4842F=\frac{48}{4^2}$
Square 4 first.
$F=4816F=\frac{48}{16}$
Divide.
$F = 3$

Common Mistake

Not squaring first

In $F=48d2F=\frac{48}{d^2}$ , you must square the distance before dividing. For $d = 4$ , use 16, not 4.

7. Recognising the graphs

The symbol $∝\propto$ means “is proportional to”.

Here are the key graph shapes:

The four key proportion graph shapes can be recognised from their overall form and position on the axes.

$\propto x$ : a straight line through the origin.
$\propto \frac{1}{x}$ : a curved graph with two branches, usually in the first and third quadrants when the constant is positive.
$\propto x^2$ : a U-shaped parabola through the origin.
$\propto \frac{1}{x^2}$ : two curved branches above the $x$ -axis when the constant is positive.

Example

Matching graph shapes

Four sketches are labelled P, Q, R, and S.

Match each sketch to the correct proportion statement.

$y$ is directly proportional to $x$ : choose P, because direct proportion gives a straight line through the origin.
$y$ is directly proportional to $x^2$ : choose Q, because squaring gives a U-shaped parabola.
$y$ is inversely proportional to $x$ : choose R, because the graph has two opposite branches.
$y$ is inversely proportional to $x^2$ : choose S, because $x^2$ is always positive, so the graph sits above the $x$ -axis.

Exam technique

In the exam

Decide first: direct means “both change the same way”; inverse means “one goes up, the other goes down”.
For word problems, find a unit rate or a total amount of work, and keep the units clear.
For formula questions, substitute carefully; if you see $x^2$ , square before you divide or multiply.

Self review

Check yourself

If 6 identical machines take 5 hours, should 3 machines take more time or less time?
What graph shape do you expect for $y = k x$ ?
In $F=80d2F=\frac{80}{d^2}$ , what should you do to $d$ before dividing 80?

Direct and Inverse Proportion

What you'll learn

1. The key idea: scaling

2. Direct proportion

Direct proportion with a formula

3. Mixed direct proportion costs

4. Inverse proportion

5. Inverse proportion formulae

6. Square and inverse square proportion

7. Recognising the graphs

Recap questions

You've reached the end

Direct and Inverse Proportion

What you'll learn

1. The key idea: scaling

2. Direct proportion

Direct proportion with a formula

3. Mixed direct proportion costs

4. Inverse proportion

5. Inverse proportion formulae

6. Square and inverse square proportion

7. Recognising the graphs

Recap questions

You've reached the end

Direct and Inverse Proportion

What you'll learn

1. The key idea: scaling

2. Direct proportion

Direct proportion with a formula

3. Mixed direct proportion costs

4. Inverse proportion

5. Inverse proportion formulae

6. Square and inverse square proportion

7. Recognising the graphs

Recap questions

You've reached the end

Direct and Inverse Proportion Revision Guide

Direct and Inverse Proportion

What you'll learn

1. The key idea: scaling

2. Direct proportion

Direct proportion with a formula

3. Mixed direct proportion costs

4. Inverse proportion

5. Inverse proportion formulae

6. Square and inverse square proportion

7. Recognising the graphs

Recap questions

You've reached the end

Direct and Inverse Proportion Revision Guide