GCSE Maths Practice: direct-proportion

Recognising Direct Proportion in Real-World Contexts

At Higher GCSE level, direct proportion questions often focus on understanding relationships rather than performing calculations. Direct proportion describes situations where two quantities increase or decrease together at a constant rate. This means that when one quantity is multiplied by a certain factor, the other quantity is multiplied by the same factor.

The most important idea behind direct proportion is the presence of a constant unit rate. This could be a fixed amount produced per machine, a constant speed, or a steady hourly pay rate. If this unit rate remains unchanged as quantities vary, the relationship is directly proportional.

How to Identify Direct Proportion

One reliable method for recognising direct proportion is to check whether doubling, tripling, or halving one quantity causes the other quantity to change by the same factor.

Example: If a printer produces 200 pages in 1 hour, it should produce 400 pages in 2 hours and 600 pages in 3 hours at the same rate. Because the output scales evenly with time, the relationship is directly proportional.

Using Unit Rate as a Check

The unit rate is the value for one unit of a quantity, such as pay per hour or distance per hour. If the unit rate stays the same across different situations, direct proportion applies.

Example: If a delivery driver earns £60 for 5 hours of work, the hourly rate is £12. If the driver earns £96 for 8 hours, the hourly rate is still £12, confirming direct proportion.

Situations That Are Not Directly Proportional

Not all relationships that involve numbers are directly proportional. If one quantity changes but the other stays fixed, or if the rate changes, then direct proportion does not apply.

• Fixed quantities that do not depend on another variable.
• Situations where the rate changes over time.
• Examples involving thresholds, bonuses, or limits.

Being able to reject these cases is just as important as identifying correct examples.

Why This Matters in Exams

Higher-tier exam questions often include plausible-sounding distractors. These are designed to test whether you understand the underlying relationship rather than relying on surface features. Carefully checking whether a constant rate exists helps avoid these traps.

Frequently Asked Questions

Does direct proportion always involve numbers?
No. It describes a relationship, which can be explained verbally as well as numerically.

Is constant speed required?
Yes, for distance–time relationships to be directly proportional, speed must remain constant.

Study Tip

In multiple-answer GCSE questions, test each option using the same rule: ask whether changing one quantity by a factor causes the other to change by the same factor. Consistency is the key sign of direct proportion.