GCSE Maths Practice: inverse-proportion

Question 8 of 10

This question tests whether you can recognise inverse proportion using algebra.

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\( \begin{array}{l} \text{Which algebraic expressions show inverse proportion?} \end{array} \)

Select all correct options:

Recognising Inverse Proportion Algebraically (Higher Tier)

This question focuses on identifying inverse proportion using algebraic expressions rather than word problems. At Higher GCSE level, you are expected to recognise inverse proportion both numerically and symbolically.

The Standard Algebraic Form

If one variable is inversely proportional to another, the relationship is written as:

y ∝ \frac{1}{x}

This means that:

y = \frac{k}{x}

where k is a constant of proportionality.

What This Means

In inverse proportion, as x increases, y decreases in such a way that the product xy stays constant. This is the defining feature of inverse proportion.

How to Check an Algebraic Expression

  1. Look for a variable in the denominator.
  2. Check whether the expression can be written in the form k ÷ x.
  3. If it can, the relationship is inverse proportion.

If the variable appears multiplied rather than divided, the relationship is not inverse proportion.

Worked Example (Not From the Options)

Example: Suppose y is inversely proportional to x and y = 8 when x = 4.

  • Use the form y = k / x
  • Substitute the values: 8 = k / 4
  • So k = 32
  • The equation is y = 32 / x

This confirms the inverse proportional relationship.

Comparing with Other Types of Relationships

It is important not to confuse inverse proportion with other common algebraic forms:

  • Direct proportion: y = kx
  • Quadratic relationships: y = x²
  • Linear but non-proportional: y = 3x + 2

Only expressions where the variable is in the denominator represent inverse proportion.

Common Higher-Tier Mistakes

  • Assuming any fraction represents inverse proportion.
  • Missing that the variable must be in the denominator.
  • Confusing inverse proportion with negative correlation.
  • Forgetting that k must be a constant.

Why This Matters in Exams

Higher GCSE exam questions often combine algebra with proportional reasoning. You may be asked to identify inverse proportion, find the value of k, or model a situation using an equation. Being confident with the form y = k / x is essential.

Study Tip

If you are unsure, rearrange the expression. If it can be written as a constant divided by a variable, it represents inverse proportion.

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