GCSE Maths Practice: direct-proportion

Understanding Direct Proportion Through Algebra and Statements

At Higher GCSE level, direct proportion questions often require you to recognise proportional relationships from equations, algebraic forms, or written statements rather than from simple numerical examples. This tests whether you understand the underlying structure of proportional relationships.

When two variables are directly proportional, they increase or decrease together at a constant rate. This relationship is written using the proportionality symbol:

y ∝ x

This can always be rewritten as an equation involving a constant of proportionality.

The Algebraic Form of Direct Proportion

If y is directly proportional to x, the relationship must take the form:

y = kx

where k is a constant number. This constant tells us how much y changes when x increases by 1. Any equation that can be written in this form represents direct proportion.

Example: If y = 7x, then when x = 1, y = 7; when x = 2, y = 14; and when x = 5, y = 35. In each case, y changes in the same ratio as x.

Using Statements to Identify Proportion

Direct proportion can also be described using words instead of equations. A common verbal description is that when x changes by a factor, y changes by the same factor.

Example: If the number of hours worked doubles and the total pay also doubles, this shows direct proportion. The key feature is that the rate of change stays constant.

Distinguishing Direct and Inverse Proportion

It is important not to confuse direct proportion with inverse proportion. In inverse proportion, one variable increases while the other decreases so that their product stays constant. This type of relationship is written as:

y = k / x

In this case, doubling x causes y to halve, which is the opposite behaviour to direct proportion.

Common Mistakes to Avoid

• Assuming any straight-line equation shows direct proportion.
• Forgetting that direct proportion graphs must pass through the origin.
• Confusing y = kx with y = k + x.
• Misinterpreting inverse relationships as direct ones.

A useful check is to divide y by x. If the result is always the same constant, the relationship is directly proportional.

Why This Skill Matters

Recognising proportional relationships algebraically is essential for higher-level mathematics, including graph interpretation, modelling real-world situations, and understanding inverse relationships. It also helps avoid common exam traps where equations look similar but represent different types of relationships.

Frequently Asked Questions

Does y = kx always represent direct proportion?
Yes. Any equation of this form represents a direct proportional relationship.

Can k be negative or fractional?
Yes. At Higher tier, k can be any non-zero number.

Study Tip

For GCSE Higher exams, always rewrite proportional statements as equations. This makes it much easier to identify whether a relationship is direct, inverse, or neither.