Who this is for: Cambridge IGCSE Mathematics (0580/0607) students who want variation — direct proportion, inverse proportion and joint variation — to become a reliable source of marks instead of a topic they only half-remember.
What query it owns: how to understand and revise variation in Cambridge IGCSE Mathematics.
Why this is safe: this page owns the variation revision-guide angle, while Tutopiya’s Variation subtopic page owns the learning resource and the free variation quiz owns the practice.

Variation describes how one quantity changes in relation to another. In Cambridge IGCSE Mathematics (0580/0607) it appears as direct proportion (y ∝ x), inverse proportion (y ∝ 1/x) and joint variation combining several variables. Examiners often wrap it in word problems about speed, workers or pressure. This guide explains the subtopic, the command words you will meet, and where to practise.

Key takeaways

• Direct variation: y = kx — as one increases, the other increases in fixed ratio.
• Inverse variation: y = k/x — as one increases, the other decreases; product k stays constant.
• Joint variation: y = kxz combines direct links to more than one variable.
• Always find k first from one pair of values, then use it to answer the question.

What is variation in Cambridge IGCSE Maths?

Variation is the algebraic description of proportion. When y varies directly as x, doubling x doubles y. When y varies inversely as x, doubling x halves y. In Cambridge IGCSE Mathematics you write y = kx or y = k/x, find the constant k from given values, and use the equation to find unknown quantities.

Study the worked examples on Tutopiya’s Variation subtopic page before attempting questions.

The core ideas you must master

Idea	What it means	How the exam uses it
Direct proportion	y = kx; y/x is constant	”y is directly proportional to x”
Inverse proportion	y = k/x; xy is constant	”y is inversely proportional to x”
Joint variation	y = kxz	”y varies directly as x and z”
Constant k	The fixed multiplier	”Find the value of y when x = 6”
Square / cube variation	y = kx² or y = k/x²	”y is directly proportional to the square of x”

How to solve a direct variation problem — step by step

1. Write the equation y = kx (or y = kx² if the question says “square of x”).
2. Substitute known values to find k.
3. Write the full equation with your value of k.
4. Substitute the new value and calculate the answer.
5. Check: does doubling x double y? If yes, direct variation is consistent.

Confirm with the free Variation quiz.

Direct vs inverse: which relationship does the question describe?

Relationship	Equation	Signal words	What stays constant
Direct	y = kx	”directly proportional”, “varies as”, “increases with”	Ratio y/x
Inverse	y = k/x	”inversely proportional”, “varies inversely”	Product xy
Joint direct	y = kxz	”varies directly as x and z”	Ratio y/(xz)

Variation in past-paper wording: command words that matter

Command word / phrase	What the question wants	Typical stem
y is directly proportional to x	Write y = kx; find k	”y is directly proportional to x. When x = 4, y = 12. Find y when x = 7.”
y is inversely proportional to x	Write y = k/x	”y is inversely proportional to x. When x = 3, y = 8. Find y when x = 6.”
y varies directly as the square of x	Write y = kx²	”y varies directly as the square of x.”
Write an equation connecting …	State the variation equation with k	”Write an equation connecting P and V.”
Work out / Find	Calculate after finding k	”Find the value of T when h = 5.”

Worked exam-style stems (how to answer the wording)

1. “y is directly proportional to x. When x = 5, y = 20. Find y when x = 8.” y = kx → 20 = 5k → k = 4 → y = 4x → when x = 8, y = 32. Reward: equation, k, final value.
2. “y is inversely proportional to x. When x = 4, y = 9. Find y when x = 6.” y = k/x → 9 = k/4 → k = 36 → y = 36/x → when x = 6, y = 6. Reward: correct k and substitution.
3. “The time, T minutes, taken to complete a task varies inversely as the number of workers, n. When n = 6, T = 40. Find T when n = 8.” T = k/n → 40 = k/6 → k = 240 → T = 240/8 = 30 minutes. Reward: translating words to y = k/x form.

Work the full set on the Algebra topical past-paper questions.

How variation connects to the wider syllabus

Variation builds on ratio skills from Number and feeds into graph questions and real-world modelling. Pair it with Linear Equations for equation practice. The Cambridge IGCSE Maths resource hub links every Algebra subtopic in one place.

Common mistakes students make

• Mixing up direct (y = kx) and inverse (y = k/x).
• Forgetting to find k before answering the final part.
• Using y = kx when the question says “square of x” — should be y = kx².
• In inverse problems, adding values instead of keeping xy constant.
• Not writing the equation when the question asks for one explicitly.

When you need more support

If variation word problems keep confusing you, retake the Variation quiz, drill the Algebra topical past-paper questions, and book a Cambridge IGCSE Maths tutor.

Frequently asked questions

What is the difference between direct and inverse variation? Direct: y = kx — both increase together. Inverse: y = k/x — one increases as the other decreases; the product xy stays fixed.

What does y ∝ x mean? It means y is directly proportional to x — there is a constant k such that y = kx.

How do I handle y varies directly as x²? Write y = kx², substitute known values to find k, then use the equation for the new value.

How do I revise variation effectively? Practise direct and inverse separately, always finding k first. Use the variation quiz to confirm each type.

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