GCSE Maths • Foundation & Higher
GCSE Proportion Revision Guide
Recognise, model and solve direct and inverse proportion problems.
Foundation
Exam practice
Worked examples
What You'll Learn
Recognise Proportional Relationships
Identify whether quantities have a constant ratio, a constant product or no proportional relationship.
Use the Unitary Method
Find the value for one unit, then scale accurately to the required number of units.
Model Direct Proportion
Find k and use y = kx to calculate unknown values in direct proportion.
Model Inverse Proportion
Find k and use y = k/x, with x non-zero, to calculate unknown values.
Interpret Proportion Graphs
Recognise a straight line through the origin for direct proportion and reciprocal curves for inverse proportion.
Check Context and Assumptions
Decide whether a real situation genuinely follows a proportional model before applying a formula.
KEY RULES AND FORMULAE Rules
Proportion Lesson Modules
Direct Proportion
Use scale factors, unit rates and y = kx to solve relationships with a constant ratio.
Inverse Proportion
Use constant products and y = k/x to solve relationships where one factor reverses the other.
Proportion Core Subtopics
Scale Factors
Use the same multiplicative change across corresponding quantities in a direct proportion problem.
Recipes and Best Buys
Scale ingredient amounts or compare unit prices only when the relationship and units are consistent.
Rates and Compound Measures
Link proportional reasoning with speed, density, pressure and other rates.
Direct-Proportion Graphs
A relationship y = kx forms a straight line through (0, 0), with gradient k.
Inverse-Proportion Graphs
The graph y = k/x is a reciprocal curve; for k > 0 its branches lie in quadrants I and III.
Algebraic Proportion with Powers
Use statements such as y ∝ x² or y ∝ 1/√x to build the correct equation.
Worked Example
Compare Direct and Inverse Models
QUESTION
Eight identical machines complete a fixed batch in 15 hours. Assuming every machine works at the same constant rate and all machines operate throughout, find the time for 12 machines. Then explain why multiplying 15 by 12/8 would be incorrect.
- Given information: The amount of work is fixed and machine rates are equal and constant, so number of machines and time are inversely proportional.
- Method: Keep the product machines × time constant.
- Find the constant: k = 8 × 15 = 120 machine-hours.
- Use 12t = 120.
- Solve: t = 120 ÷ 12 = 10.
- Sense check: the machine count increases by a factor of 12/8 = 1.5, so time should be divided by 1.5; 15 ÷ 1.5 = 10.
- The batch takes 10 hours.
- 8 × 15 = 120 and 12 × 10 = 120, so the product remains constant.
- Multiplying the time by 12/8 treats the relationship as direct proportion. More identical machines should reduce, not increase, the completion time under the stated assumptions.
- Common GCSE Mistakes
- Choosing by Direction Alone
- Assuming Every Straight Line Is Direct Proportion
- Using the Wrong Constant
- Applying the Same Scale Factor in an Inverse Model
- Ignoring Units
- Forgetting Context Assumptions
- Exam Tips
- Identify the relationship: look for a constant ratio for direct proportion or a constant product for inverse proportion.
- Write the model before substituting: y = kx, y = k/x or the exact power relationship given.
- Show how you found k and include its units when they help explain the context.
- Use a sense check: in a direct model both quantities scale together; in an inverse model one scales in the opposite way.
- Substitute the result into the original relationship and confirm the constant is unchanged.
- State relevant assumptions in practical questions, especially fixed work, fixed distance and constant individual rates.
Ready to Practice?
Printable Worksheet
Practise unitary methods, constants of proportionality, direct and inverse models, graphs and contextual assumptions.
Interactive Quiz
Check whether relationships are direct, inverse or non-proportional and select the correct equation.
Exam-Style Questions
Apply proportion to algebraic, graphical and practical multi-step GCSE contexts.
Watch the Proportion Tutorial
Clear, step-by-step explanation of expanding brackets, simplifying complex expressions, and solving for x. Perfect for visual learners who need a breakdown of the rules in action.
- 12 minute comprehensive guide
- High-definition whiteboard walk-through
Continue Your GCSE Maths Revision
Continue Your GCSE Maths Revision
Frequently Asked Questions
Proportion describes a multiplicative relationship between quantities. Directly proportional quantities have a constant ratio, while inversely proportional quantities have a constant product. Not every relationship in which two values change is proportional.
For direct proportion y = kx, divide y by x to find k. For inverse proportion y = k/x, multiply x by y. Substitute another pair of values to check that the same constant is obtained.
In direct proportion, equal scale factors act in the same direction and y/x stays constant. In inverse proportion, one factor is reversed and xy stays constant. Direction alone is not proof, so test the ratio or product.
If y = kx and x = 0, then y = 0. Therefore every direct-proportion graph contains (0, 0). A straight line with a non-zero y-intercept may show a linear relationship, but it is not direct proportion.
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