Direct and Inverse Proportion (Variation)
Proportion questions all follow the same four steps, whatever the power involved. Once you can write the relationship correctly, the rest is substitution — and nearly every lost mark comes from the first step.
1. Writing the relationship
∝ means “is proportional to”
| In words | Statement | Formula |
|---|---|---|
| y is directly proportional to x | y ∝ x | y = kx |
| y is proportional to x² | y ∝ x² | y = kx² |
| y is proportional to x³ | y ∝ x³ | y = kx³ |
| y is proportional to √x | y ∝ √x | y = k√x |
| y is inversely proportional to x | y ∝ 1/x | y = k/x |
| y is inversely proportional to x² | y ∝ 1/x² | y = k/x² |
| y is inversely proportional to √x | y ∝ 1/√x | y = k/√x |
“Inversely” means the variable goes on the BOTTOM. That is the entire difference, and getting it the wrong way round was the most common error recorded.
The power applies to the variable, not to k. For y ∝ x², the formula is y = kx² — not (kx)² and not y = k²x. Forgetting to square x entirely was also recorded.
Reading the wording carefully: “proportional to the square of x” means x²; “proportional to the square root of x” means √x. Misreading one for the other changes the whole answer.
2. The four-step method
1. Write the formula with k. 2. Substitute the given pair to find k. 3. Rewrite the formula with k in it. 4. Substitute the new value.
Example: y is directly proportional to x². When x = 3, y = 18. Find y when x = 5.
- y = kx²
- 18 = k(3²) = 9k → k = 2
- y = 2x²
- y = 2(5²) = 50
Example: y is inversely proportional to x. When x = 4, y = 5. Find y when x = 10.
- y = k/x
- 5 = k/4 → k = 20
- y = 20/x
- y = 20/10 = 2
Step 3 is worth writing out explicitly. The question often asks you to “find the formula connecting x and y” as a separate part, and it earns its own mark.
Recalculate k for each new relationship. Carrying a value of k over from an earlier part of a question, when the relationship has changed, was a specific recorded error. Each proportionality has its own constant.
3. Recognising which type
Direct proportion: as one goes up, the other goes up. The graph is a straight line through the origin (for y ∝ x).
Inverse proportion: as one goes up, the other goes down. The graph is a reciprocal curve, and xy = k is constant.
Sanity-check the direction of your answer. For inverse proportion, a larger x must give a smaller y. If your answer moves the wrong way, you have used the wrong relationship — this check catches the most common error instantly.
Typical contexts:
- Direct: cost and quantity; distance and time at constant speed; circumference and radius
- Inverse: speed and time for a fixed journey; number of workers and time taken; pressure and volume
4. Working backwards
You may be given y and asked for x.
Example: y = 2x², find x when y = 72.
- 72 = 2x² → x² = 36 → x = ±6 (take the positive value if the context requires it)
Remember the ± when square-rooting, unless the context — a length, a number of people — rules out the negative.
5. Proportion in geometry
Similar shapes are a proportion question in disguise:
Area ∝ (length)² and Volume ∝ (length)³
So if lengths double, area is ×4 and volume ×8. The same four-step method applies with k as the constant of proportionality.
Check whether a geometry question is really about proportion. Not recognising this was recorded in lessons — questions about scaling models, or a volume given a length, are proportion problems.
6. Scope note
One lesson covered the inverse square law in a physics context.
The tutor correctly noted this is enrichment, not syllabus. For 0580 you need direct and inverse variation with powers of 1, 2, 3 and square roots — check your own syllabus document.
7. Mistakes that cost marks
Confusing direct with inverse proportion.
Putting the variable on the wrong side of the fraction.
Forgetting to square or cube the variable.
Misreading “square” as “square root”.
Applying the power to k instead of the variable.
Reusing k from an earlier, different relationship.
Not writing out the completed formula.
Forgetting ± when square-rooting.
Not checking the direction of the answer.
Frequently asked questions
What does ∝ mean? “Is proportional to.”
What is direct proportion? As one quantity increases, the other increases in the same ratio: y = kx.
What is inverse proportion? As one increases, the other decreases: y = k/x, with xy constant.
How do I write “y is inversely proportional to x squared”? y = k/x².
How do I find k? Substitute the given pair of values into the formula.
What are the four steps? Write the formula with k; find k; rewrite the formula; substitute the new value.
Do I use the same k throughout? Only within the same relationship. A new relationship needs a new k.
How do I check my answer? For direct proportion both values move the same way; for inverse they move in opposite directions.
What does the graph look like? Direct (y ∝ x): a straight line through the origin. Inverse: a reciprocal curve.
Quick revision checklist
- I know what ∝ means
- I can write direct and inverse statements as formulae
- I know “inversely” puts the variable on the bottom
- I apply powers to the variable, not to k
- I read “square” and “square root” carefully
- I use the four-step method
- I write out the completed formula
- I calculate a fresh k for each new relationship
- I can work backwards to find x from y
- I include ± where appropriate
- I check the direction of my answer
- I recognise proportion in similar-shape questions
These notes cover direct and inverse proportion in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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