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MasterMaths online tuition: GCSE Maths Direct and Inverse Proportion Explained

GCSE Maths Direct and Inverse Proportion Explained

Writer: Ryan Harvey
Ryan Harvey
Oct 12
3 min read

Direct and inverse proportion connect quantities through a constant relationship. The words describe how one variable changes when another changes, but the algebraic formulas are different. GCSE questions often require students to find the constant of proportionality from one set of values and then use that constant to find another. Learning what changes and what stays fixed prevents confusion.


Recognise direct proportion

If y is directly proportional to x, write y = kx, where k is a constant. Doubling x doubles y while other conditions remain unchanged. If y = 3 when x = 6, then 3 = 6k so k = 1/2. The rule is y = x/2. For x = 10, y = 5. A direct-proportion graph of y against x is a straight line passing through the origin, not just any straight line.


Recognise inverse proportion

If y is inversely proportional to x, write y = k/x for nonzero x. Doubling positive x halves y. If y = 10 when x = 2, then k = xy = 20, so y = 20/x. When x = 5, y = 4. The product xy stays constant, whereas the ratio y/x stays constant in direct proportion. Writing this distinction beside a worked example often prevents the wrong formula being chosen.


Handle proportion to a square

If y is directly proportional to x², then y = kx². Suppose y = 18 when x = 3. Substituting gives 18 = 9k, so k = 2 and y = 2x². When x = 4, y = 32. Doubling x in this model multiplies y by four, not two. Pay attention to the exact wording: “directly proportional to the square” is different from “directly proportional to x”.


Inverse proportion to a square

For y inversely proportional to x², use y = k/x². If y = 8 when x = 2, then k = yx² = 8 × 4 = 32. Thus y = 32/x² and, for x = 4, y = 2. These variations are common in Higher questions because they assess careful translation of language into algebra. Check whether the relevant quantity is x, x² or another expression before substituting numbers.


Solve a real-world question

Imagine the time taken by a fixed workforce is inversely proportional to the number of equally productive workers. If four workers take six hours, the constant product is 24 worker-hours. Six workers would take four hours under the simplified assumptions. Real projects may not behave exactly this way, so the model must be understood as conditional. Mathematical proportionality questions normally assume other variables are fixed unless stated otherwise.


Read graphs and check results

Direct-proportion graphs through the origin grow as x grows when k is positive. Inverse-proportion graphs for positive k and positive x curve downwards without crossing the axes. Check the relationship with a known pair and substitute it back into your final expression. A frequent mistake is to say quantities are directly proportional just because one increases when the other increases; the ratio must remain constant.


Frequently asked questions

Is every increasing relationship direct proportion?

No. Direct proportion requires a constant ratio and a graph through the origin.

What remains constant in inverse proportion?

For y = k/x, the product xy is constant.

Related GCSE revision resources

The existing guides below explore supporting skills. Choose the explanations that match the area where further understanding is required.

Book a free online maths introduction

If this topic remains challenging, discuss whether individual online maths tuition would help. MasterMaths offers a free 20-minute introductory conversation with Ryan. Bring your student’s year group, exam board and a relevant question. The objective is a clear learning plan rather than a promise of a particular grade.

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