What does the symbol mean?

"Is proportional to". It is always replaced by "= k " to make an equation you can work with.

R10, R13Ratio, Proportion & Rates of ChangeFoundation & HigherGrades 3–8

Direct & Inverse Proportion

Q: Is every straight-line graph direct proportion?

No — only straight lines through the origin. y = 2x + 3 is linear but not proportional, because doubling x does not double y.

Two quantities are in direct proportion when doubling one doubles the other — like the cost of petrol and the litres bought. They are in inverse proportion when doubling one halves the other — like the number of workers and the time a job takes. Recognising which is which, and using one-unit reasoning, solves a huge range of real problems.

45 min Video by Maths Genie AQA GCSE Maths

The big picture

For direct proportion, the ratio $\frac{y}{x}$ stays constant, so $y = kx$ and the graph is a straight line through the origin. For inverse proportion, the product $xy$ stays constant, so $y = \frac{k}{x}$ and the graph is a curve that never touches the axes.

At Higher tier, proportion extends to powers and roots: the distance a ball falls is proportional to the square of the time, $d = kt^2$ . The method never changes — write the formula with $k$ , find $k$ from a known pair, then use it.

Direct and Inverse ProportionCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Solve direct proportion problems with the unitary method, including best buys
Solve inverse proportion problems
Recognise direct and inverse proportion from tables and graphs
Form and use equations such as $y = kx^2$ and $y = \frac{k}{\sqrt{x}}$ (Higher)

Direct proportion: the unitary method

Find the value for one unit, then multiply up. Keep units and labels in your working so the steps are clear.

For best buys, compare either the price per gram (or ml) or the amount per penny. The better value is the lower price per unit.

Example 16 bread rolls cost £2.10. How much do 15 rolls cost?

1One roll:

2.10 \div 6 = £0.35

215 rolls:

15 \times 0.35 = £5.25

Answer

£5.25

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

8 litres of paint cost £36. How much do 5 litres cost, in pounds?

Your answer

Show worked solution

Solution

1One litre: $36 \div 8 = £4.50$ .
25 litres: $5 \times 4.50 = £22.50$ .

Answer: £22.50

Example 2Which is better value: 400 g of cereal for £2.80, or 750 g for £4.95?

1400 g:

280 \div 400 = 0.7

p per gram.

2750 g:

495 \div 750 = 0.66

p per gram.

Answer

The 750 g box is better value (0.66p per gram).

Inverse proportion

If one quantity is multiplied by a number, the other is divided by it. Find the "total" first — such as worker-days or person-hours — then divide.

Example 14 builders take 15 days to build a wall. How long would 6 builders take, working at the same rate?

1Total work:

4 \times 15 = 60

builder-days.

26 builders:

60 \div 6 = 10

days.

Answer

10 days

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

A pump empties a pond in 12 hours. How many hours would 3 identical pumps take?

Your answer

Show worked solution

Solution

1Three times as many pumps take a third of the time.
2 $12 \div 3 = 4$ hours.

Answer: $4$ hours

Tip — Ask "should the answer be bigger or smaller?" More builders means fewer days — if your answer goes up, the proportion is inverse and you have multiplied instead of dividing.

Recognising proportion from tables and graphs

Direct proportion: $\frac{y}{x}$ is the same for every pair, and the graph is a straight line through $(0, 0)$ .

Inverse proportion: $x \times y$ is the same for every pair, and the graph is a curve $y = \frac{k}{x}$ .

Direct proportion is a straight line through the origin; inverse proportion is a curve that approaches both axes.

Example 1Is

y

directly or inversely proportional to

x

x: 2, 4, 5, 10

and

y: 30, 15, 12, 6

1Products:

2 \times 30 = 60

4 \times 15 = 60

5 \times 12 = 60

10 \times 6 = 60

2The product is constant.

Answer

Inversely proportional:

y = \frac{60}{x}

Algebraic proportion (Higher)

" $y$ is proportional to $x$ " is written $y \propto x$ , and becomes $y = kx$ . The same pattern works for any power or root: $y \propto x^2$ gives $y = kx^2$ ; $y \propto \frac{1}{\sqrt{x}}$ gives $y = \frac{k}{\sqrt{x}}$ .

Method: (1) write the equation with $k$ , (2) substitute the known pair to find $k$ , (3) write the full formula, (4) use it.

Example 1

y

is directly proportional to

x^2

. When

x = 4

y = 40

. Find

y

when

x = 6

y = kx^2

40 = k \times 16

, so

k = 2.5

and

y = 2.5x^2

3When

x = 6

y = 2.5 \times 36

Answer

y = 90

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

$d$ is directly proportional to $t^2$ . When $t = 3$ , $d = 45$ . Find $d$ when $t = 5$ .

Your answer

Show worked solution

Solution

1 $d = kt^2$ : $45 = 9k$ , so $k = 5$ .
2 $d = 5 \times 25 = 125$ .

Answer: $d = 125$

Example 2

F

is inversely proportional to the square of

r

. When

r = 2

F = 50

. Find

F

when

r = 5

F = \frac{k}{r^2}

50 = \frac{k}{4}

, so

k = 200

F = \frac{200}{25}

Answer

F = 8

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

$y$ is inversely proportional to $\sqrt{x}$ . When $x = 9$ , $y = 8$ . Find $y$ when $x = 36$ .

Your answer

Show hint

$y = \frac{k}{\sqrt{x}}$ , and $\sqrt{9} = 3$ .

Show worked solution

Solution

1 $8 = \frac{k}{3}$ , so $k = 24$ .
2When $x = 36$ : $y = \frac{24}{6} = 4$ .

Answer: $y = 4$

Proportion as scale factors

Scale factors give a shortcut. If $y \propto x^2$ and $x$ is multiplied by 3, then $y$ is multiplied by $3^2 = 9$ . If $y \propto \frac{1}{x^2}$ and $x$ is doubled, $y$ is divided by 4. These "what happens if…" questions appear regularly at grades 7–9.

Think like an examiner

For best buys, the comparison needs figures for every option and a conclusion.
Write the formula with the value of $k$ (e.g. $y = 2.5x^2$ ) — "find a formula" questions award a mark for it.
Check for the words "square", "cube" and "root" — missing them is the most common way to lose all the marks.

Proportion

y \propto x \Rightarrow y = kx

Direct: straight line through the origin.

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

Inverse: product

xy

is constant.

y \propto x^n \Rightarrow y = kx^n

Find

k

from a known pair.

Watch out for these

Treating workers and time as direct proportion: 6 builders take

15 \times 1.5 = 22.5

days.

More workers, less time:

60 \div 6 = 10

days.

Writing

y \propto x^2

y = kx

Include the power:

y = kx^2

Comparing best buys by total price only.

Compare price per unit amount.

y \propto x^2

and

x

doubles, saying

y

doubles.

y

is multiplied by

2^2 = 4

Stretch yourself

The time $T$ for a pendulum swing is proportional to the square root of its length $L$ . A pendulum 64 cm long takes 1.6 s. How long must a pendulum be to take 2 s?

Hint — Write $T = k\sqrt{L}$ , find $k$ , then rearrange for $L$ .

Reveal solution

Solution

1.6 = k\sqrt{64} = 8k

, so

k = 0.2

and

T = 0.2\sqrt{L}

. For

T = 2

\sqrt{L} = 10

, so

L = 100

cm. (Scale-factor check: time

\times 1.25

means length

\times 1.25^2 = 1.5625

, and

64 \times 1.5625 = 100

✓.)

\blacksquare

Questions students ask

What does the symbol

\propto

mean?

"Is proportional to". It is always replaced by "

= k \times

" to make an equation you can work with.

Is every straight-line graph direct proportion?

No — only straight lines through the origin.

y = 2x + 3

is linear but not proportional, because doubling

x

does not double

y

How do I decide between direct and inverse?

Ask what happens when one quantity increases. If the other increases in the same ratio, it is direct; if it decreases, it is inverse.

Key takeaways

Unitary method: find one unit, then scale.
Best buy: compare price per unit.
Inverse: find the constant product, then divide.
Direct graphs are straight through the origin; inverse graphs are $\frac{k}{x}$ curves.
Higher: write $y = kx^n$ , find $k$ , then use the formula.

Exam-style questions

Practise it the way it is examined

3 original questions · 10 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 33 marks

A recipe for 4 people uses 300 g of rice and 450 ml of stock.

How much rice and stock are needed for 10 people?[3]
Show mark scheme
- M1Finds the amount for one person, e.g. 75 g rice, or uses a multiplier of 2.5
- A1750 g rice
- A11125 ml stock

Question 2Grade 53 marks

It takes 5 people 9 hours to paint a hall. The hall must be painted in 6 hours.

How many extra people are needed, assuming everyone works at the same rate?[3]
Show mark scheme
- M1Total $5 \times 9 = 45$ person-hours
- M1 $45 \div 6 = 7.5$ , so 8 people needed
- A13 extra people
Answer: 3

Examiner note

7.5 people is not possible — round up to 8, then subtract the 5 already working.

Question 3Grade 74 marks

$P$ is inversely proportional to the cube of $q$ . When $q = 2$ , $P = 12.5$ .

(a)Find a formula for $P$ in terms of $q$ .[2]
Show mark scheme (a)
- M1 $P = \frac{k}{q^3}$ and $12.5 = \frac{k}{8}$
- A1 $P = \frac{100}{q^3}$
Answer: $P = \dfrac{100}{q^3}$
(b)Find $q$ when $P = 0.8$ .[2]
Show mark scheme (b)
- M1 $q^3 = \frac{100}{0.8} = 125$
- A1 $q = 5$
Answer: $q = 5$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

3 pens cost £1.20. How much do 7 pens cost?
- £2.80
- £3.60
- £2.40
- £8.40
Show answer and explanation
Answer: £2.80
One pen costs 40p.
2 people take 6 hours. How long do 3 people take?
- 4 hours
- 9 hours
- 3 hours
- 6 hours
Show answer and explanation
Answer: 4 hours
12 person-hours ÷ 3.
Which graph shows direct proportion?
- A straight line through the origin
- Any straight line
- A curve through the origin
- A horizontal line
Show answer and explanation
Answer: A straight line through the origin
$y = kx$ passes through $(0, 0)$ .
Better value: 500 ml for £1.20 or 1.5 litres for £3.30?
- 1.5 litres for £3.30
- 500 ml for £1.20
- The same
- Cannot tell
Show answer and explanation
Answer: 1.5 litres for £3.30
0.24p/ml vs 0.22p/ml.
$y = kx$ and $y = 21$ when $x = 3$ . Find $y$ when $x = 5$ .
- $35$
- $23$
- $7$
- $105$
Show answer and explanation
Answer: $35$
$k = 7$ .
$y \propto \frac{1}{x}$ and $y = 8$ when $x = 3$ . Find $y$ when $x = 12$ .
- $2$
- $32$
- $4$
- $96$
Show answer and explanation
Answer: $2$
$k = 24$ , $y = 24 \div 12$ .
$y \propto x^2$ . If $x$ is multiplied by 3, $y$ is multiplied by…
- $9$
- $3$
- $6$
- $27$
Show answer and explanation
Answer: $9$
$3^2 = 9$ .
$y \propto \frac{1}{x^2}$ and $y = 20$ when $x = 1$ . Find $y$ when $x = 2$ .
- $5$
- $10$
- $80$
- $40$
Show answer and explanation
Answer: $5$
$y = \frac{20}{4}$ .

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