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Direct and inverse proportion

Free Printable Direct and inverse proportion PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 5)

Download free direct and inverse proportion worksheets (PDF), GCSE maths past paper questions and worked solutions for Direct and inverse proportion (5B). This is a Grade 5 ratio & proportion topic from the GCSE questions by topic collection for Foundation and Higher tier crossover GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Maths4Everyone, Maths Made Easy, Dr Frost, Dr Austin, 1st Class Maths, Metatutor, GCSE Maths Rocket, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our direct and inverse proportion revision notes and worked examples with diagrams, try real GCSE past paper direct and inverse proportion questions with step-by-step solutions, or download a PDF of all direct and inverse proportion worksheet links.

Free Direct and inverse proportion PDF Worksheets & Answers

Grade 5 • Topic 5B
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Grade 5 topics
5A - Combining ratios5B - Direct and inverse proportion
5C - Reverse percentages5D - Standard form
5E - Speed5F - Changing the subject
5G - Quadratic expressions5H - Solving quadratic equations
5J - Drawing quadratic graphs5K - Cubic/reciprocal graphs
5L - Simultaneous equations5M - Sector areas and arc lengths
5N - Sim. equ. graphically5P - Midpoint of a line segment
5Q - Gradient of a line5R - Equation of a line
5S - Spheres and cones5T - Similar shapes (lengths)
5U - Trigonometry (SOHCAHTOA)5V - Exact trig. values
5W - Probability trees5X - Vectors
5Y - Venn diagrams5Z - Density
5AA - Pressure
GCSE Maths Grade 5 Revision Notes & Method

Guide to Direct and inverse proportion: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

If y is directly proportional to x (written y ∝ x), then y = kx. If y is inversely proportional to x (written y ∝ 1x), then y = kx. Read the question carefully for powers or roots such as x2, x3, or √x.

Always follow three steps: (1) write the proportion equation with constant k, (2) substitute the given pair of values to find k and write the full formula, and (3) substitute the new value to find the unknown.

Example 1: y is directly proportional to x2. When x = 3, y = 45. Find y when x = 5.

Solution:

Write equation: y = kx2
Substitute x = 3, y = 45: 45 = k × 32 → 45 = 9k → k = 5
Formula: y = 5x2
When x = 5: y = 5 × 52 = 5 × 25 = 125
Answer: y = 125

Example 2: y is inversely proportional to x. When x = 4, y = 15. Find x when y = 12.

Solution:

Write equation: y = kx
Substitute x = 4, y = 15: 15 = k4 → k = 15 × 4 = 60
Formula: y = 60x
When y = 12: 12 = 60x → x = 60 ÷ 12 = 5
Answer: x = 5
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Direct and inverse proportion Questions & Answers

Past Paper Example 1 OCR Nov 2023 4H Q8 2 marks
Full Paper (PDF) • Mark Scheme (PDF)
Here are sketches of six graphs, labelled **A** to **F**.
AyxOByxOCyxODyxOEyxOFyxO
(a) y is directly proportional to x. [1 mark]
(b) y is inversely proportional to x. [1 mark]
Worked Solution & Mark Scheme (2 marks)
Part (a) Solution:
Direct proportion y = kx is a straight line through the origin ⇒ Graph C. B1
Answer (a): C
Part (b) Solution:
Inverse proportion y = k/x is a reciprocal curve in the first quadrant ⇒ Graph E. B1
Answer (b): E
Past Paper Example 2 OCR Nov 2018 3F Q24 / 6H Q9 2 marks
Full Paper (PDF) • Mark Scheme (PDF)
These graphs show different relationships between the variables x and y.
Graph AOxyGraph BOxyGraph COxyGraph DOxy
(a) Identify the graph which shows the following.
y is directly proportional to x. [1 mark]
(b) y is inversely proportional to x. [1 mark]
Worked Solution & Mark Scheme (2 marks)
Part (a) Solution:
Direct proportion y = kx is a straight line through the origin ⇒ Graph A. B1
Answer (a): A
Part (b) Solution:
Inverse proportion y = k/x is a reciprocal hyperbola ⇒ Graph C. B1
Answer (b): C
Past Paper Example 3 IGCSE Jun 2018 2H Q16 5 marks
Full Paper (PDF) • Mark Scheme (PDF)
R is proportional to t2
The graph shows the relationship between R and t for 0 ≤ t ≤ 4
123410203040ORt
(a) Find a formula for R in terms of t. [3 marks]
(b) show that t is inversely proportional to √(x) for t > 0 [2 marks]
Worked Solution & Mark Scheme (5 marks)
Part (a) Solution:
R = kt² M2
at (2, 10), 10 = 4k ⇒ k = 2.5 ⇒ R = 2.5t² A1
Answer (a): R = R = 2.5t2
Part (b) Solution:
(5/2)t² = 8/(5x) M1
⇒ t² = 16/(25x)
⇒ t = (4/5)/√x = 0.8/√x C1
Answer (b): 2.5t² = 8/(5x) ⇒ t² = 16/(25x) ⇒ t = 0.8/√x, so t is inversely proportional to √x

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