If 1 apple costs £2, how much do 4 apples cost?
Multiply by 4.
Cost increases directly. 2 × 4 = £8.
Direct Proportion
Direct proportion describes a relationship where two quantities increase or decrease together at the same rate. This is often written in the form y = kx.
Overview
Two quantities are in direct proportion when they increase or decrease in the same ratio.
If one value doubles, the other also doubles. If one triples, the other also triples.
\( y \propto x \qquad \text{and} \qquad y = kx \)
The constant k is called the constant of proportionality. This topic appears in tables, recipes, cost problems, speed-style contexts and algebra questions.
What you should understand after this topic
- Understand what direct proportion means
- Recognise direct proportion from words, tables and equations
- Find the constant of proportionality k
- Use y = kx to solve missing values
- Understand how direct proportion differs from inverse proportion
Key Definitions
Direct Proportion
Two quantities change in the same ratio.
Proportional To
The symbol \( \propto \) means “is proportional to”.
Constant of Proportionality
The fixed number \(k\) in the equation \( y = kx \).
Multiplier
The number used to scale one quantity to another.
Unit Value
The amount for 1 unit, often used to solve proportion tables.
Linear Relationship Through the Origin
A direct proportion graph is a straight line that passes through \( (0,0) \).
Key Rules
Double one, double the other
If \(x\) doubles, \(y\) doubles.
Use \( y = kx \)
This is the standard direct proportion equation.
Find \(k\) by dividing
\( k = \frac{y}{x} \)
Graph passes through origin
A direct proportion graph goes through \( (0,0) \).
Quick Recognition
Words
“is directly proportional to”
Equation
\( y = 4x \)
Table pattern
Multiply by the same scale factor each time.
Graph pattern
Straight line through the origin.
How to Solve
What does direct proportion mean?
Direct proportion means two variables are linked by a constant multiplier. It is based on ideas from ratio, where one quantity scales in proportion to another.As one increases, the other increases in the same ratio.
\( y \propto x \quad \Longrightarrow \quad y = kx \)
If \( y \) is directly proportional to \( x \), then \( \frac{y}{x} \) is constant.
Exam tip: Direct proportion always produces a straight line through the origin.
Step 1: Turn proportion into an equation
If a question says:
\( y \text{ is directly proportional to } x \)
\( y = kx \)
The constant \(k\) is called the constant of proportionality.
Step 2: Find the constant of proportionality
Use given values to calculate \(k\).
If \( y = 18 \) when \( x = 6 \):
\( 18 = k \cdot 6 \)
Divide both sides by 6: \( k = 3 \).
So the equation becomes \( y = 3x \).
Exam tip: Always find \(k\) before answering the question.
Step 3: Use the equation
Once you know \(k\), substitute new values.
If \( x = 10 \):
\( y = 3 \times 10 = 30 \)
Alternative method (unitary method)
You can also solve direct proportion using the unitary method.
Find the value for 1 unit first.
Then scale up to the required value.
This method is closely related to best value and everyday calculations.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on identifying and solving problems involving direct proportion.
Edexcel
Identify whether \( y \) is directly proportional to \( x \) in the equation \( y = 5x \).
Edexcel
Given that \( y \propto x \) and \( y = 12 \) when \( x = 3 \), find \( y \) when \( x = 10 \).
Edexcel
Given that \( y \propto x \), and \( y = 18 \) when \( x = 6 \), find the constant of proportionality.
Edexcel
The cost of 4 notebooks is £10. Assuming the cost is directly proportional to the number of notebooks, find the cost of 9 notebooks.
Edexcel
The distance travelled is directly proportional to time. If a car travels 120 km in 2 hours, how far will it travel in 5 hours at the same speed?
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on forming and solving equations involving direct proportion.
AQA
Given that \( y \propto x \), and \( y = 20 \) when \( x = 4 \), find an equation connecting \( y \) and \( x \).
AQA
If \( y \propto x \), and \( y = 15 \) when \( x = 5 \), find \( x \) when \( y = 27 \).
AQA
The mass of an object is directly proportional to its volume. If the mass is 24 kg when the volume is 3 m^3, find the mass when the volume is 7 m^3.
AQA
The circumference of a circle is directly proportional to its diameter. Given that the circumference is 31.4 cm when the diameter is 10 cm, find the circumference when the diameter is 15 cm.
AQA
Explain how you recognise a direct proportion from a table or graph.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising reasoning, graphical interpretation, and algebraic modelling.
OCR
Given that \( y \propto x \), and \( y = 8 \) when \( x = 2 \), find the value of \( y \) when \( x = 9 \).
OCR
Write \( y \propto x \) as an equation using a constant of proportionality.
OCR
A worker is paid £45 for 5 hours of work. Assuming pay is directly proportional to time worked, find the pay for 8 hours.
OCR
A recipe uses 200 g of flour to make 8 pancakes. How much flour is needed to make 14 pancakes?
OCR
State the key features of a graph that represents direct proportion.
Exam Checklist
Step 1
If you see “directly proportional”, write \( y = kx \).
Step 2
Substitute a known pair of values to find \( k \).
Step 3
Use the completed equation to find the missing value.
Step 4
For graphs, check that the line passes through the origin.
Most common exam mistakes
Wrong formula
Using \( y = k + x \) instead of \( y = kx \).
Wrong constant
Not dividing correctly to find \( k \).
Graph confusion
Thinking any straight line is direct proportion.
Context mistake
Forgetting to keep units sensible in money, mass or length questions.
Common Mistakes
These are common mistakes students make when working with direct proportion in GCSE Maths.
Not starting with the correct form
Incorrect
A student tries to solve the problem without writing an equation.
Correct
Always start with the form \( y = kx \), where k is the constant of proportionality.
Using addition instead of multiplication
Incorrect
A student adds a constant instead of multiplying by one.
Correct
In direct proportion, values are multiplied by a constant factor. The relationship must follow \( y = kx \), not \( y = x + c \).
Finding k incorrectly
Incorrect
A student divides or multiplies the wrong values when calculating k.
Correct
To find k, divide y by x using known values. For example, \( k = \frac{y}{x} \).
Assuming any straight line shows direct proportion
Incorrect
A student thinks all linear graphs represent direct proportion.
Correct
Only lines that pass through the origin represent direct proportion. Other straight lines follow \( y = mx + c \).
Not checking the origin
Incorrect
A student accepts a graph without checking if it passes through (0,0).
Correct
A key feature of direct proportion is that the graph passes through the origin. Always check this before concluding.
Confusing with inverse proportion
Incorrect
A student uses inverse proportion methods for a direct proportion problem.
Correct
Direct proportion has the form \( y = kx \), while inverse proportion follows \( y = \frac{k}{x} \). Make sure you identify the correct relationship.
Try It Yourself
Practise solving problems involving direct proportional relationships.
Foundation
Foundation Practice
Solve direct proportion problems using scaling and simple multiplication.
3 books cost £12. How much do 5 books cost?
Find cost of 1 book first.
1 book = £4 → 5 books = £20.
A car travels 60 km in 1 hour. How far does it travel in 3 hours?
Multiply distance by 3.
60 × 3 = 180 km.
5 pencils cost £10. How much do 2 pencils cost?
Find cost of 1 pencil first.
1 pencil = £2 → 2 pencils = £4.
If 2 kg of apples cost £6, how much does 6 kg cost?
Multiply by 3.
2 kg → £6, so 6 kg → £18.
8 metres of fabric cost £24. How much do 3 metres cost?
Find cost per metre.
1 m = £3 → 3 m = £9.
A student says if 4 items cost £8, then 8 items cost £12. What is wrong?
Doubling items should double cost.
If items double, cost must double. £8 → £16, not £12.
7 litres of fuel cost £14. How much do 10 litres cost?
Find cost per litre.
1 litre = £2 → 10 litres = £20.
If y is directly proportional to x, what happens when x doubles?
Direct proportion means same scale factor.
If x doubles, y also doubles.
6 oranges cost £3. How much do 15 oranges cost?
Find cost of 1 orange.
1 orange = £0.50 → 15 = £7.50.
Higher
Higher Practice
Solve direct proportion problems using equations and constant k.
y is directly proportional to x. When x = 4, y = 12. Find y when x = 10.
Find k first.
y = kx → 12 = 4k → k = 3 → y = 3 × 10 = 30.
y is directly proportional to x. When x = 5, y = 20. Find y when x = 8.
Find k using y = kx.
k = 4 → y = 4 × 8 = 32.
y ∝ x. If y = 18 when x = 6, what is the value of k?
Use y = kx.
18 = 6k → k = 3.
y ∝ x. When x = 3, y = 15. Find y when x = 7.
Find k first.
k = 5 → y = 5 × 7 = 35.
A quantity y is directly proportional to x. When x increases by a factor of 4, what happens to y?
Same multiplier applies.
Direct proportion means same factor applies to y.
y ∝ x. When x = 2, y = 10. Find x when y = 25.
Find k first, then rearrange.
k = 5 → y = 5x → 25 = 5x → x = 5.
A student says y = x + 3 shows direct proportion. Why are they wrong?
Direct proportion must pass through origin.
Direct proportion must be y = kx. This has a constant term.
y ∝ x. When x = 8, y = 24. Find x when y = 15.
Find k first.
k = 3 → y = 3x → 15 = 3x → x = 5.
Which equation represents direct proportion?
Must be y = kx.
Only y = 4x matches y = kx.
y ∝ x. When x = 6, y = 9. Find y when x = 20.
Find k first.
k = 1.5 → y = 1.5 × 20 = 30.
Games
Practise this topic with interactive games.
Games coming soon.