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Cambridge IGCSE International Maths · Extended
Proportion
The big idea: Proportion is just a rulebook for how two quantities move together —
either they grow together in lockstep (direct), or one grows while the other shrinks
to compensate (inverse). Once you find the "magic number" k that links them,
you can predict either value from the other, forever.
Summary — What This Chapter Covers
Direct proportion (y ∝ x): as x increases, y increases by the same factor. Equation: y = kx. Graph: a straight line through the origin.
Direct proportion with powers/roots: y can be proportional to x², x³, √x, or ∛x — not just x itself.
Inverse proportion (y ∝ 1/x): as x increases, y decreases by the same factor. Equation: y = k/x. Graph: a curve that never touches the axes.
Inverse proportion with powers/roots: y can be proportional to 1/x², 1/x³, 1/√x, or 1/∛x.
The constant of proportionality, k, is found by substituting one known pair of values into the general formula.
The 4-step method — identify variables, form the equation with k, solve for k, rewrite and use — works for every single proportion question, direct or inverse.
Variation models (linear, quadratic, cubic, reciprocal) let you match real data to the correct type of equation, and test/compare which model fits best.
1. Direct Proportion
What does "directly proportional" actually mean? Imagine you're paying for fuel. If 1 litre costs $2, then 2 litres cost $4, and 10 litres cost $20. Every time the amount of fuel doubles, the price doubles too. The ratio between the two quantities (price ÷ litres) never changes — it's always $2 per litre. That's direct proportion in a nutshell: two quantities that scale up and down together, always keeping the same ratio.
We write "y is directly proportional to x" using the special symbol ∝, like this: y ∝ x. This symbol is just shorthand — it doesn't give you a usable equation on its own. To actually calculate anything, you need to convert it into an equation using a constant, k:
Core Formula — Direct Proportion
y ∝ x ⟶ y = kx
"y equals some fixed number k, multiplied by x." The graph of this is a straight line through the origin (0,0), and k is literally the gradient of that line.
Here's what that line looks like — notice it always passes through the origin, which makes sense: if x = 0 (you buy 0 litres of fuel), y must be 0 too (you pay $0).
y
| ●
| ●
| ●
| ●
|●________________________ x
0
A straight line through the origin.
Steeper line = bigger k.
Why does this matter?
Because k is CONSTANT, once you know it, you can predict y for ANY value of x — even ones you haven't been given. That's the whole power of proportion: one piece of data (one known x-y pair) unlocks the entire relationship.
1Direct Proportion with Powers and Roots
Not every direct proportion is as simple as y = kx. Sometimes y is proportional to a power or root of x instead of x itself. The idea is exactly the same — just swap out the "x" for whatever power/root you're told about:
y proportional to the square of x → y ∝ x² → y = kx²
y proportional to the square root of x → y ∝ √x → y = k√x
y proportional to the cube of x → y ∝ x³ → y = kx³
y proportional to the cube root of x → y ∝ ∛x → y = k∛x
Each of these produces a different shaped graph — a squared relationship gives a curve that starts flat and rockets upward; a square-root relationship gives a curve that shoots up quickly then flattens out. You don't need to memorise every shape, but you should recognise that "proportional to x²" is not the same graph as "proportional to x."
2The 4-Step Method for Direct Proportion
Every single direct proportion question — no matter how it's dressed up — is solved with the exact same four steps. Memorise this process and you can answer almost any proportion question on the exam:
Step 1 — Identify & Formula: Work out which two variables are involved and write the general formula with k in it (e.g. y = kx², y = k√x, etc.)
Step 2 — Find k: Substitute the one pair of values you're given into the formula, then solve the resulting equation for k.
Step 3 — Rewrite: Plug your value of k back into the formula. Now you have the FULL, specific equation linking the two variables.
Step 4 — Use it: Substitute in whatever new value the question asks about, to find the missing answer.
Worked Example
y is directly proportional to the square of x. When x = 3, y = 18. Find y when x = 4.
Step 1: "Square of x" → y = kx²
Step 2: Substitute x = 3, y = 18 → 18 = k(3)² → 18 = 9k → k = 2
Step 3: Full equation: y = 2x²
Step 4: Substitute x = 4 → y = 2(4)² = 2 × 16 = 32
Practice Question 1
y is directly proportional to x. When x = 5, y = 20. Find the value of y when x = 8.
Practice Question 2
p is directly proportional to the cube of q. When q = 2, p = 40. Find p when q = 3.
2. Inverse Proportion
Now flip the idea on its head. Imagine 4 people painting a fence together take 6 hours. If you bring in more painters, the job goes faster — the time goes DOWN as the number of people goes UP. That's inverse proportion: as one variable increases, the other decreases by the same factor, so their product stays constant (not their ratio, like direct proportion).
Here's the clever trick to understanding inverse proportion: it's actually just direct proportion to the reciprocal of the other variable. "y is inversely proportional to x" really means "y is directly proportional to 1/x." We write this as:
Core Formula — Inverse Proportion
y ∝ 1/x ⟶ y = k/x
"y equals a fixed number k, divided by x." As x gets bigger, k/x gets smaller (because you're dividing k by a bigger number) — that's exactly the "opposite movement" behaviour we want.
The graph of y = k/x looks completely different from direct proportion — it's a curve (called a hyperbola) that gets closer and closer to both axes but never actually touches them:
y
|●
| ●
| ●
| ●●
| ●●●●●●●●●●●____ x
0
Curve drops steeply, then flattens.
Never touches either axis.
Common Mistake
Students often write y = kx by accident when the question says "inversely proportional" — the word "inversely" is easy to skim past under exam pressure. Train yourself: the moment you see "inversely," your hand should automatically write k over x, never k times x.
1Inverse Proportion with Powers and Roots
Just like direct proportion, inverse proportion can involve a power or root of x. The pattern is identical to before — just put the power/root underneath k instead of next to it:
y inversely proportional to the square of x → y ∝ 1/x² → y = k/x²
y inversely proportional to the square root of x → y ∝ 1/√x → y = k/√x
y inversely proportional to the cube of x → y ∝ 1/x³ → y = k/x³
y inversely proportional to the cube root of x → y ∝ 1/∛x → y = k/∛x
2The 4-Step Method for Inverse Proportion
Exactly the same four steps as direct proportion — the only thing that changes is the shape of your formula in Step 1 (k is now on top, divided by x, not multiplied by x):
Step 1 — Identify & Formula: Write the general formula, e.g. y = k/x, y = k/x², y = k/∛x
Step 2 — Find k: Substitute the known pair of values, then solve for k.
Step 3 — Rewrite: Substitute k back in to get the full, specific equation.
Step 4 — Use it: Substitute the new given value to find the answer.
Worked Example — Full Multi-Part Question
The time, t hours, to complete a project is inversely proportional to the cube root of the number, n, of people working on it. If 27 people work on the project, it takes 50 hours.
(a) Find an equation connecting t and n.
Step 1: "Inversely proportional to cube root of n" → t = k/∛n
Step 2: Substitute n = 27, t = 50 → 50 = k/∛27 = k/3 → k = 150
Step 3: Full equation: t = 150/∛n
(b) The project must finish within 60 hours. Find the minimum number of people needed.
Since n must be a whole number of people, and 15 people would be too slow, round up to 16 people.
Exam-Winning Insight
Part (b) of that example is a classic exam trap: the maths gives you 15.625, but you can't have 0.625 of a person! Always ask "does this answer need to make real-world sense?" — and remember whether you should round up or down based on the context (here, fewer people = slower, so you must round UP to guarantee the deadline is met).
Practice Question 3
y is inversely proportional to x. When x = 6, y = 5. Find y when x = 2.
Practice Question 4
The pressure, P, of a gas is inversely proportional to the square of its volume, V. When V = 4, P = 10. Find P when V = 2.
3. Variation Models
A "variation model" is just a formal name for the equation that describes how two quantities relate. You've already met four of them — you just didn't call them "models" yet:
y = kx → linear model (this IS direct proportion) — straight line graph
y = kx² → quadratic model — U-shaped / parabola-like curve
y = kx³ → cubic model — S-shaped curve
y = k/x → reciprocal model (this IS inverse proportion) — hyperbola curve
The new skill in this section isn't a new formula — it's detective work: given a table of raw data (with no equation given), can you figure out WHICH of these models actually fits? And if you're given two or more competing models, can you test them against the data to see which one is more accurate?
The Strategy in One Sentence
Try the simplest model first (linear), find k using ONE data point, then check if that same k correctly predicts the OTHER data points. If it doesn't fit, move to the next model (quadratic, then cubic) and repeat.
1Finding Missing Values in a Table
Sometimes you're told exactly which type of proportion applies (e.g. "y is inversely proportional to x") and given a table with some gaps. This is just the standard 4-step method — but instead of stopping after finding one missing value, you use the same final equation to fill in every gap in the table.
Worked Example
Given that y is inversely proportional to x, find p and q:
x
1
3
q
y
36
p
9
Step 1: y = k/x
Step 2: Use the ONE complete pair (x=1, y=36): 36 = k/1 → k = 36
This is the trickiest skill in the chapter, so let's slow right down. You'll be given a table of data with NO equation and asked to work out which model (linear, quadratic, cubic, or reciprocal) fits best. The method is systematic — don't guess, test:
Pick a candidate model (start with the simplest, y = kx)
Use the first data pair to calculate k
Use that same k to predict the y-value for the second x-value
Compare your prediction to the ACTUAL y-value in the table
If they match → you've found the right model! Double check with a third point to be sure.
If they don't match → the model is wrong. Try the next one (quadratic, then cubic).
Worked Example
x
2
4
6
y
20
80
180
Try linear (y = kx): Using x=2,y=20 → k=10. Predict x=4: y=10×4=40. Actual is 80. ✗ Doesn't fit — reject.
The best model is quadratic, with full equation y = 5x².
Worked Example — Comparing Two Suggested Models
Model A: y = 400/x² Model B: y = 800/x³. Which fits the data better?
x
5
6
10
15
Actual y
25
15
3
2
Model A
16
11.1
4
1.78
Model B
6.4
3.7
0.8
0.24
Compare each model's predictions to the actual values. Model A's numbers (16, 11.1, 4, 1.78) are much closer to the real data (25, 15, 3, 2) than Model B's numbers are. Model A is the better fit.
Practice Question 5
A table shows: when x = 2, y = 6; when x = 3, y = 13.5. Given the relationship is quadratic (y = kx²), find k and predict y when x = 5.
What to Memorise
Direct Proportion
y ∝ x means y = kx. As x rises, y rises by the same factor. Graph = straight line through origin, gradient = k.
Inverse Proportion
y ∝ 1/x means y = k/x. As x rises, y falls by the same factor. Graph = curve approaching but never touching both axes.
Constant of Proportionality (k)
The fixed number linking the two variables. Found by substituting ONE known pair of x,y values into the general formula and solving.
The ∝ Symbol
Means "is proportional to." It is NOT an equation on its own — always convert it to an equals sign with k before calculating anything.
Direct with Power/Root
y ∝ xⁿ → y = kxⁿ (works for squares, cubes, square roots, cube roots — n can be any power or root).
Inverse with Power/Root
y ∝ 1/xⁿ → y = k/xⁿ (the power or root goes in the denominator).
The 4-Step Method
1) Identify variables & write formula with k → 2) Substitute known pair to find k → 3) Rewrite full equation → 4) Use equation to answer the question.
Variation Model
A general term for the equation type describing a relationship: linear (kx), quadratic (kx²), cubic (kx³), or reciprocal (k/x).
Testing a Model Against Data
Find k using ONE data point, then use that k to predict at least one OTHER data point. If the prediction matches the actual value, the model fits.
Concepts Checklist
Exam Tips & Common Mistakes
Mistake: Forgetting to Find k First
Never try to jump straight from "y ∝ x²" to an answer. You cannot use the equation until you know k — always find it first using the given pair of values, even if the question doesn't explicitly say "find the equation."
Mistake: Mixing Up Direct and Inverse
"Directly proportional" → multiply by k (y = kx). "Inversely proportional" → divide by k... no wait, divide k BY x (y = k/x). Read the word "inversely" twice before you commit to a formula — this single word flips your whole equation.
Tip: Powers Belong to x, Not k
If y ∝ x², the square applies to x, never to k. Write y = kx² — NOT y = (kx)² or k²x. This is one of the most common algebra slips in this topic.
Tip: Always Show the "Full Equation" Step
Even if the exam question only asks for a final numerical answer, examiners award method marks for writing out the complete equation (Step 3) before substituting the new value. Never skip straight from finding k to the final answer — write the equation in between.
Tip: Sanity-Check Real-World Answers
If your variable represents something that must be a whole number (like people, or items), and your calculation gives a decimal, think carefully about whether to round up or down based on what the question is actually asking — don't just round to the nearest whole number automatically.
Mistake: Testing a Model with Only One Data Point
Finding k from ONE point and stopping there proves nothing — any two points can always be connected by SOME value of k. You must check that the SAME k correctly predicts at least one other point in the table before concluding the model fits.