GCSE Maths Practice: direct-proportion

Question 8 of 10

This question tests your ability to recognise direct proportion using algebra.

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\( \begin{array}{l}\text{If } y \propto x \text{ and } y = 10 \text{ when } x = 4, \text{ which pairs are true?}\end{array} \)

Select all correct options:

Divide y by x to check whether the constant of proportionality stays the same.

Direct Proportion Expressed Algebraically

At Higher GCSE level, direct proportion questions often move beyond numerical scaling and require algebraic reasoning. When two variables are directly proportional, this relationship can be written using algebra, which allows you to test whether different pairs of values follow the same proportional rule.

The symbol ∝ means “is proportional to”. If y is directly proportional to x, we write:

y ∝ x

This relationship can be rewritten as an equation by introducing a constant of proportionality.

The Constant of Proportionality

When y is directly proportional to x, the relationship always has the form:

y = kx

The value k is called the constant of proportionality. It represents how much y increases for each increase of 1 in x. Once k is known, the relationship between x and y is completely defined.

Example: If y = 12 when x = 3, then k = 12 ÷ 3 = 4. This gives the equation y = 4x. Any valid pair must satisfy this equation.

How to Check Value Pairs

To check whether a pair of values follows a direct proportion rule, substitute the x-value into the equation y = kx and see whether the resulting y-value matches the given one.

This method is more reliable than trying to scale numbers mentally, especially when values are not simple multiples. It is the standard approach expected in Higher-tier exam questions.

Common Mistakes to Avoid

  • Forgetting to calculate the constant of proportionality first.
  • Assuming all pairs that “look close” are correct.
  • Confusing direct proportion with linear relationships that have an added constant.
  • Using y = x instead of y = kx.

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

Why Algebra Matters Here

Using algebra makes proportional reasoning precise and consistent. It allows you to test any value pair, even when numbers are large or awkward. This skill is also essential for later topics such as graphs of proportional relationships and inverse proportion.

Frequently Asked Questions

Is y = kx always a straight line?
Yes. It produces a straight line passing through the origin when graphed.

Can k be a decimal or fraction?
Yes. At Higher tier, k is often non-integer to test accuracy and reasoning.

Study Tip

For GCSE Higher exams, always rewrite direct proportion statements as equations. This avoids guesswork and ensures every pair is checked using the same logical method.

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