J560 · 2.03, 5.01–5.03Ratio & ProportionFoundation & HigherGrades 5–8

Direct & Inverse Proportion

Q: What does the symbol ∝ mean?

"Is proportional to". Replace it with "= k " to form an equation.

If 4 pens cost £3.20, then 10 pens cost £8 — doubling the pens doubles the cost. That is $\textbf{direct proportion}$ . If 4 workers take 12 days to build a wall, then 6 workers take 8 days — more workers, less time. That is $\textbf{inverse proportion}$ . At Higher tier, both extend to squares, cubes and roots, and are written as equations with a constant $k$ .

50 min Ratio & Proportion

The big picture

Direct proportion means one quantity is a constant multiple of the other: $y = kx$ . Inverse proportion means their product is constant: $y = \dfrac{k}{x}$ . Recognising which applies, then finding $k$ from one pair of values, turns any proportion question into substitution.

OCR Higher questions often start " $y$ is inversely proportional to the square of $x$ ", and the marks go to writing the correct equation, finding $k$ , and using it. Graphs also appear: direct proportion is a straight line through the origin; inverse proportion is a reciprocal curve.

GCSE Maths - What Does Inversely Proportional Mean? (2026/27 exams)Curated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

Before you start

Try a short calculation from the preceding topic without the notes.
Check the tier and grade band shown in this lesson.

Watch, pause, try

Video by Cognito. The video’s own board label is retained; follow the specification reference and scope notes on this lesson for assessed content.

Viewing focus: Solve direct and inverse proportion problems using unitary and multiplier methods

After watching, attempt these questions before opening the answers. You can also use the written lesson below.

After watching 1: 5 kg of apples cost £7.50. How much do 8 kg cost?

Show answer and reasoning

£12. £1.50 per kg.

After watching 2: 3 pumps empty a tank in 10 hours. How long do 5 pumps take?

Show answer and reasoning

6 hours. 30 pump-hours ÷ 5.

What you'll learn

Solve direct and inverse proportion problems using unitary and multiplier methods
Recognise direct and inverse proportion from tables and graphs
Form equations such as $y = kx^{2}$ and $y = \dfrac{k}{\sqrt{x}}$ and use them (Higher)
Describe the effect of changing one variable on another

Direct proportion

Two quantities are in direct proportion if multiplying one by a number multiplies the other by the same number. Their ratio stays constant.

The unitary method finds the value for one unit: if 4 pens cost £3.20, one costs 80p, so 10 cost £8.

As an equation, $y \propto x$ means $y = kx$ . Its graph is a straight line through the origin with gradient $k$ .

y \propto x \iff y = kx

Direct proportion.

Your turnSame method, new numbers — try it before moving on.

5 kg of potatoes cost £12.50. How much do 8 kg cost, in pounds?

Your answer

Inverse proportion

Two quantities are inversely proportional if multiplying one by a number divides the other by the same number. Their product stays constant.

If 4 workers take 12 days, the job is $4 \times 12 = 48$ worker-days, so 6 workers take $48 \div 6 = 8$ days.

As an equation, $y \propto \dfrac{1}{x}$ means $y = \dfrac{k}{x}$ . Its graph is a reciprocal curve.

y \propto \tfrac{1}{x} \iff y = \tfrac{k}{x}

Inverse proportion.

Your turnSame method, new numbers — try it before moving on.

6 workers take 10 days to build a wall. How many days would 4 workers take?

Your answer

Why this works

Direct proportion keeps the $\textit{ratio}$ constant; inverse proportion keeps the $\textit{product}$ constant. Checking which of $\tfrac{y}{x}$ or $xy$ stays the same in a table identifies the relationship immediately.

Proportion with powers and roots

" $y$ is directly proportional to the square of $x$ " means $y = kx^{2}$ . " $y$ is inversely proportional to the square root of $x$ " means $y = \dfrac{k}{\sqrt{x}}$ .

The method is always: write the equation with $k$ , substitute the given pair to find $k$ , write the equation with the value of $k$ , then use it.

Example 1

y

is directly proportional to

x^{2}

. When

x = 3

y = 18

. Find

y

when

x = 5

y = kx^{2}

, so

18 = 9k

and

k = 2

y = 2x^{2}

3When

x = 5

y = 2 \times 25

Answer

y = 50

Your turnSame method, new numbers — try it before moving on.

$y$ is directly proportional to $x^3$ . When $x = 2$ , $y = 24$ . Find $y$ when $x = 3$ .

Your answer

Example 2

y

is inversely proportional to

\sqrt{x}

. When

x = 4

y = 6

. Find

y

when

x = 9

y = \dfrac{k}{\sqrt{x}}

, so

6 = \dfrac{k}{2}

and

k = 12

2When

x = 9

y = \dfrac{12}{3}

Answer

y = 4

Tip — Write the equation with $k$ found as a separate line, e.g. " $y = 2x^{2}$ ". Mark schemes award a mark for it.

The effect of a change

If $y = kx^{2}$ and $x$ is doubled, $y$ is multiplied by $2^{2} = 4$ . If $y = \dfrac{k}{x^{2}}$ and $x$ is doubled, $y$ is divided by 4.

For a percentage change, use multipliers: if $x$ increases by 20% in $y = kx^{3}$ , $y$ is multiplied by $1.2^{3} = 1.728$ , a 72.8% increase.

Example 1The force

F

between two magnets is inversely proportional to the square of the distance

d

between them. The distance increases by 20%. By what percentage does the force change?

F = \dfrac{k}{d^{2}}

; new distance

1.2d

2New force

= \dfrac{k}{1.44d^{2}} = 0.694F

AnswerThe force decreases by about 30.6%.

Your turnSame method, new numbers — try it before moving on.

$y$ is directly proportional to $x^2$ . $x$ is increased by 20%. By what percentage does $y$ increase?

Your answer

Think like an examiner

Write the proportionality equation with $k$ before substituting.
Show the value of $k$ and the equation with $k$ replaced.
Use multipliers for percentage changes and explain the conclusion.

Proportion

y = kx

Direct proportion.

y = \tfrac{k}{x}

Inverse proportion.

y = kx^{n},\ y = \tfrac{k}{x^{n}}

With powers or roots.

Watch out for these

Writing "

y

inversely proportional to

x^{2}

" as

y = kx^{2}

y = \dfrac{k}{x^{2}}

Assuming

y

doubles when

x

doubles in

y = kx^{2}

It quadruples.

Using

\tfrac{y}{x}

to test inverse proportion.

For inverse proportion,

xy

is constant.

Forgetting to square

x

when finding

k

Substitute into the full equation:

18 = k \times 3^{2}

Stretch yourself

The time $t$ for a pendulum to swing is directly proportional to the square root of its length $l$ . A pendulum of length 64 cm takes 1.6 s. Find the time for a pendulum of length 1 m, and the length needed for a time of 2.4 s.

Hint — Use consistent units. Form $t = k\sqrt{l}$ with $l$ in cm.

Questions students ask

How do I tell direct from inverse proportion?

Direct: both increase together and the ratio is constant. Inverse: one increases as the other decreases and the product is constant.

What does the symbol ∝ mean?

"Is proportional to". Replace it with "

= k \times

" to form an equation.

Is every straight-line graph direct proportion?

No — only straight lines through the origin.

Key takeaways

Direct proportion: $y = kx$ , constant ratio, straight line through the origin.
Inverse proportion: $y = \tfrac{k}{x}$ , constant product, reciprocal curve.
With powers and roots, write the equation, find $k$ , then substitute.
Doubling $x$ in $y = kx^{n}$ multiplies $y$ by $2^{n}$ .
Use multipliers to find the effect of percentage changes.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 53 marks

$d$ is directly proportional to $t^2$ . When $t = 2$ , $d = 20$ .

(a)Find a formula for $d$ in terms of $t$ .[2]
(b)Find $d$ when $t = 5$ .[1]

Question 2Grade 72 marks

$P$ is inversely proportional to $V$ . When $V = 40$ , $P = 150$ . Find $V$ when $P = 250$ .[2]

Question 3Grade 83 marks

$y$ is inversely proportional to $x^2$ . $x$ is increased by 25%. Find the percentage decrease in $y$ .[3]

Unlimited practice

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OCR J560 official specification (opens in a new tab)
Check the assessed scope and practical or mathematical requirements.
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An external explanation or interactive activity to use alongside the lesson. It may include extension material.

Return to this tomorrow

Close the notes and answer from memory. Revisit any explanation you cannot yet reconstruct.

$y \propto x$ and $y = 21$ when $x = 3$ . Find $y$ when $x = 10$ .
$y$ is inversely proportional to $x$ . When $x = 5$ , $y = 8$ . Find $y$ when $x = 20$ .
$y = kx^{3}$ . If $x$ is halved, $y$ is:
$P$ is inversely proportional to $V$ . $P = 12$ when $V = 3$ . Which is the equation?

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

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Related lessons

All 5 lessons

Direct & Inverse Proportion

If 4 pens cost £3.20, then 10 pens cost £8 — doubling the pens doubles the cost. That is

\textbf{direct proportion}

. If 4 workers take 12 days to build a wall, then 6 workers take 8 days — more workers, less time. That is

\textbf{inverse proportion}

. At Higher tier, both extend to squares, cubes and roots, and are written as equations with a constant

k

50 min Ratio & Proportion

Watch, pause, try

Video by Cognito. The video’s own board label is retained; follow the specification reference and scope notes on this lesson for assessed content.

Viewing focus: Solve direct and inverse proportion problems using unitary and multiplier methods

After watching, attempt these questions before opening the answers. You can also use the written lesson below.

After watching 1: 5 kg of apples cost £7.50. How much do 8 kg cost?

Show answer and reasoning

£12. £1.50 per kg.

After watching 2: 3 pumps empty a tank in 10 hours. How long do 5 pumps take?

Show answer and reasoning

6 hours. 30 pump-hours ÷ 5.

Practise it the way it is examined

Question 1Grade 53 marks

$d$ is directly proportional to $t^2$ . When $t = 2$ , $d = 20$ .

(a)Find a formula for $d$ in terms of $t$ .[2]
(b)Find $d$ when $t = 5$ .[1]

Question 2Grade 72 marks

$P$ is inversely proportional to $V$ . When $V = 40$ , $P = 150$ . Find $V$ when $P = 250$ .[2]

Question 3Grade 83 marks

$y$ is inversely proportional to $x^2$ . $x$ is increased by 25%. Find the percentage decrease in $y$ .[3]

Return to this tomorrow

Close the notes and answer from memory. Revisit any explanation you cannot yet reconstruct.

y \propto x

and

y = 21

when

x = 3

. Find

y

when

x = 10

y

is inversely proportional to

x

. When

x = 5

y = 8

. Find

y

when

x = 20

y = kx^{3}

. If

x

is halved,

y

is:

P

is inversely proportional to

V

P = 12

when

V = 3

. Which is the equation?