GCSE Maths Practice: inverse-proportion

Solving Inverse Proportion Problems Using Algebra (Higher Tier)

This question tests a key Higher GCSE Maths skill: solving inverse proportion problems using algebra. You are expected to recognise the relationship, form the correct equation, and use substitution to find missing values.

The Inverse Proportion Formula

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

y ∝ \frac{1}{x}

which leads to the equation:

y = \frac{k}{x}

Here, k is the constant of proportionality.

Why the Constant Matters

The constant k links all values of x and y. Even though x and y change, the product x × y remains the same. Finding k first is essential in any inverse proportion calculation.

Step-by-Step Method

1. Write down the inverse proportion formula.
2. Substitute the known values of x and y.
3. Solve to find the constant k.
4. Substitute the new value of x.
5. Calculate the corresponding value of y.

This method ensures accuracy and is the approach examiners expect to see.

Worked Example (Different Values)

Example: y is inversely proportional to x. When x = 5, y = 18. Find y when x = 9.

• y = k / x
• 18 = k / 5 → k = 90
• y = 90 / 9
• y = 10

The product x × y stays constant.

Another Worked Example

Example: y is inversely proportional to x. When x = 12, y = 7. Find y when x = 4.

• y = k / x
• 7 = k / 12 → k = 84
• y = 84 / 4
• y = 21

Common Higher-Tier Mistakes

• Forgetting to calculate the constant k.
• Using y = kx instead of y = k / x.
• Making arithmetic errors when dividing.
• Not checking whether the answer makes sense.

Why This Skill Is Important

Inverse proportion appears frequently in Higher GCSE exams, often combined with algebra, rearranging formulas, or multi-step reasoning. Mastering this technique helps you handle more complex proportional relationships confidently.

Study Tip

Always write down the formula first. If the relationship is inverse, the variable must be in the denominator.