Direct Proportion

SPEC R10 • RATIO, PROPORTION & RATES OF CHANGE

Writing, finding and using y = kx — and its Higher Tier extensions to squares, cubes and roots.

Why This Topic Matters

Direct proportion problems are a reliable 3–4 mark question across all three papers. They test whether you can read the phrasing of the relationship carefully, set up an equation with an unknown constant, find that constant, and then use the full relationship to answer the question.

The Big Idea

"y is directly proportional to x" is a code phrase. Translated, it means y = kx for some fixed number k that you must find from the information given. Every proportion question follows the same three-step pattern.

The Three Steps — always the same

Write the relationship using a constant k (e.g. y = kx, y = kx², y = k√x).

Find k by substituting the pair of values you are given.

Apply the full relationship to answer the actual question.

What Direct Proportion Means

Definition

Direct proportion. Two quantities are in direct proportion if one is always a constant multiple of the other. Written "y ∝ x", which means y = kx for a fixed constant k, called the constant of proportionality.

The Two Defining Behaviours

As x doubles, y doubles. As x triples, y triples. The ratio y : x stays constant.

When x = 0, y = 0. The graph must pass through the origin.

Example of direct proportion: the total cost of petrol is directly proportional to the number of litres bought — double the litres, double the cost, and 0 litres costs £0.

Not direct proportion: a taxi fare with a £3 fixed charge plus £2 per mile. When you travel 0 miles the cost is £3, not £0, so the graph of cost against miles does not pass through the origin. The two quantities are not in direct proportion, even though the graph is a straight line.

Spotting Direct Proportion in a Table

If y is directly proportional to x, then the ratio y ÷ x is the same value (k) for every pair in the table. Check this with a quick division.

x25812

y6152436

y ÷ x3333

Reading the Table

Every pair gives y ÷ x = 3. So k = 3 and the relationship is y = 3x. That single equation generates every row of the table — and every pair you haven't yet seen.

Quick Counter-Check

If one pair gave y ÷ x = 3 and another gave y ÷ x = 4, the quantities would not be in direct proportion — even if they looked similar.

The Graph of y = kx

The graph of direct proportion is always a straight line through the origin with gradient k. Anything else isn't direct proportion — no matter how neat the line looks.

The Test — must satisfy BOTH

Straight line (not curved).

Passes through the origin (0, 0).

Interpreting k as a Rate of Change

When x and y have units, the constant k is a rate of change — the amount of y per one unit of x. If cost C (in £) is directly proportional to number of litres L and the graph has gradient 1.45, then the petrol costs £1.45 per litre. On a distance-time graph through the origin, the gradient is the speed.

The MCE Procedure for Proportion

Every direct proportion question is scored using the same three marks. Using the MCE framework keeps the method visible to the examiner — which is where most of the marks are.

Method

Translate the words into an equation with k. "y proportional to x" → y = kx.

Calculate

Substitute the given pair to find k. Then write the full relationship with k replaced by its value.

Evaluate

Substitute the new x-value into the full relationship and state the answer clearly, with units if relevant.

The Trap

Students often finish at the end of step C — they find k and stop. k is not the answer. The answer is the new value of y. Always take the third step.

Worked Example — Linear Direct Proportion01

y is directly proportional to x. When x = 4, y = 28. Find the value of y when x = 9. [3 marks]

M

"Directly proportional" means y = kx. I will find k from the given pair, then use the full relationship.

C

Substitute x = 4, y = 28 into y = kx: 28 = 4k → k = 7.

Full relationship: y = 7x.

E

When x = 9: y = 7 × 9 = 63. Answer: y = 63.

Ratio Method — Alternative Approach

For the linear case (y = kx only), you can skip finding k and use ratios directly. This is often quicker on a non-calculator paper and is the method most Foundation students use.

The Ratio Rule

If y is directly proportional to x, then for any two pairs (x₁, y₁) and (x₂, y₂):y₁ / x₁ = y₂ / x₂

Same Problem, Ratio Method

y is directly proportional to x. When x = 4, y = 28. Find y when x = 9.

Write the equal ratios: 28 / 4 = y / 9.

Cross-multiply: 28 × 9 = 4 × y → 252 = 4y → y = 63.

Same answer as the MCE method — both score full marks.

When to Use Which Method

The ratio method works for y = kx only. If the question mentions "proportional to the square", "the cube", or "the square root" of x, always use the MCE (k) method — the simple ratio shortcut does not apply to powers.

Higher Tier: Proportional to a Power

At Higher Tier, "y is proportional to x" is just the simplest version. The relationship can involve any power of x. The wording tells you which one.

Higher Tier Forms

Phrase in the questionEquation to writey is proportional to xy = kxy is proportional to x² (the square of x)y = kx²y is proportional to x³ (the cube of x)y = kx³y is proportional to √x (the square root of x)y = k√x

Read Every Word

Students who see "proportional" and immediately write y = kx lose every mark when the question says "proportional to the square". Read the exact phrase, then write the correct equation.

Graph Shapes by Power

The power of x in the relationship changes the shape of the graph — but all direct proportion graphs still pass through the origin.

Key Take-away

All three curves pass through the origin — that's the signature of direct proportion. The shape between x = 0 and x getting large depends on the power.

Worked Example — Proportional to a Square [Higher]02

y is directly proportional to the square of x. When x = 4, y = 80. Find the value of y when x = 7. [4 marks]

M

"Proportional to the square of x" means y = kx². Find k from the given pair, then use the full relationship.

C

Substitute x = 4, y = 80 into y = kx²: 80 = k × 16 → k = 5.

Full relationship: y = 5x².

E

When x = 7: y = 5 × 7² = 5 × 49 = 245. Answer: y = 245.

Worked Example — Proportional to a Root [Higher]03

The time T seconds taken by a pendulum to swing is directly proportional to the square root of its length L metres. A pendulum of length 0.25 m takes 1 s to swing. Find the time taken by a pendulum of length 4 m. [4 marks]

M

"T proportional to √L" means T = k√L. Find k using the given pair (L = 0.25, T = 1), then use the full relationship.

C

Substitute: 1 = k × √0.25 = k × 0.5 → k = 2.

Full relationship: T = 2√L.

E

When L = 4: T = 2 × √4 = 2 × 2 = 4. Answer: T = 4 seconds.

Common Errors to Avoid

These four mistakes account for the majority of dropped marks on proportion questions. Learn them so you don't repeat them.

Students often writeCorrect versionWhy it matters

y = kx for every proportion questionRead the wording — may be y = kx², y = kx³ or y = k√xUsing the wrong power loses every mark

Find k, then stop and circle k as the answerSubstitute the new x-value into the full relationshipk is an intermediate value, not the final answer

"As x doubles, y doubles" for y = kx²As x doubles, y is multiplied by 2² = 4Power affects scaling behaviour

Direct proportion graph with y-intercept ≠ 0Direct proportion graphs must pass through (0, 0)A y-intercept means it's not direct proportion

Finds k but leaves the equation as "y = kx²"Rewrite the equation with k's value — e.g. "y = 5x²"The full relationship is the M1/A1 step before applying

Examiner Alert

Examiner Alert — "k found but not used"

One of the most frequent mark losses on direct proportion questions: students set up y = kx, find the correct value of k, and then write k as the final answer. k is the constant of proportionality, not the answer. The question asks for a value of y (or T, or whatever variable). Always perform step 3 — substitute the new x-value into the full relationship.

Examiner Alert — Missing the Power

"y is directly proportional to the square of x" is not the same as "y is directly proportional to x." The first gives y = kx²; the second gives y = kx. Read the full sentence before writing the equation — not just the words "proportional to".

How to Show Full Working

For full marks, write all three lines of working — equation with k, value of k, full relationship. Even if you can do it in your head, the method marks are only awarded for work that is on the page.

Practice Question 1

y is directly proportional to x. When x = 6, y = 27. Find the value of y when x = 10. [3 marks]

M

Direct proportion: y = kx. Find k, then apply the full relationship.

C

27 = k × 6 → k = 27 ÷ 6 = 4.5. So y = 4.5x.

E

When x = 10: y = 4.5 × 10 = 45.

Practice Question 2 [Higher]

A is directly proportional to the square of r. When r = 3, A = 45. Find A when r = 8. [4 marks]

M

"A proportional to the square of r" means A = kr².

C

Substitute r = 3, A = 45: 45 = k × 9 → k = 5. So A = 5r².

E

When r = 8: A = 5 × 64 = 320.

Practice Question 3

Two students plot graphs to show direct proportion between y and x. Student A's graph is a straight line passing through (0, 2) with gradient 3. Student B's graph is a straight line passing through the origin with gradient 3.

(a) Which student's graph shows direct proportion? [1 mark]

(b) Give the equation of the graph that shows direct proportion. [1 mark]

(c) Explain why the other graph does not show direct proportion. [2 marks]

(a)

Student B.

(b)

y = 3x (straight line through origin with gradient 3).

(c)

Student A's graph does not pass through the origin — it has equation y = 3x + 2. For direct proportion, when x = 0 we need y = 0. A y-intercept of 2 means the ratio y ÷ x is not constant, so the quantities are not directly proportional.

Practice Question 4 [Higher]

d is directly proportional to the cube of t. When t = 2, d = 24.

(a) Find an equation for d in terms of t. [2 marks]

(b) Work out the value of d when t = 5. [2 marks]

(c) Work out the value of t when d = 375. [2 marks]

(a)

d = kt³. Substitute t = 2, d = 24: 24 = k × 8 → k = 3. d = 3t³.

(b)

d = 3 × 5³ = 3 × 125 = 375.

(c)

375 = 3t³ → t³ = 125 → t = 5 (the cube root of 125).

Practice Question 5 — Worded Context

The cost C pounds of printing a booklet is directly proportional to the number of pages n. It costs £1.80 to print a 24-page booklet.

(a) Find an equation connecting C and n. [2 marks]

(b) How much will it cost to print a 60-page booklet? [1 mark]

(c) A booklet costs £6.15 to print. How many pages does it have? [2 marks]

(a)

C = kn. Substitute n = 24, C = 1.80: 1.80 = 24k → k = 0.075. So C = 0.075n (i.e. 7.5p per page).

(b)

When n = 60: C = 0.075 × 60 = £4.50.

(c)

6.15 = 0.075 × n → n = 6.15 ÷ 0.075 = 82 pages.

Summary

The Core Idea

Direct proportion: y ∝ x means y = kx for a constant k. Graph is a straight line through the origin.

Higher Tier extensions: y ∝ x² → y = kx²; y ∝ x³ → y = kx³; y ∝ √x → y = k√x.

Key test: ratio y ÷ x (or y ÷ x², etc.) is constant for every pair.

The Three-Step Procedure (MCE)

1

M — Write the relationship using k (e.g. y = kx²).

2

C — Find k by substituting the given pair. Rewrite the full relationship with k replaced.

3

E — Apply it : substitute the new value and state the final answer.

The Three Traps

Missing the power: "proportional to the square" is y = kx², not y = kx.

k is not the answer: always carry out step three — k is the intermediate, y is the answer.

No origin, no direct proportion: any graph with a non-zero y-intercept is not direct proportion.

Graph Shapes (at a Glance)

RelationshipGraph shape

y = kxStraight line through origin

y = kx²Parabola branch from origin

y = kx³Cubic branch from origin (rises faster)

y = k√xRoot curve from origin (rises then flattens)