At a glance
- Cambridge 0580
- Extended E2.8; Core C1.11 ratio only
- Edexcel 4MA1
- Higher 2.5A; Foundation 1.7C, D
- Direct
- y = kx^n
- Inverse
- y = k/x^n
What each tier expects
| Board and tier | Reference | What can be asked |
|---|---|---|
| Cambridge 0580 Core | C1.11 | Ratio and proportional reasoning: recipes, map scales, best value |
| Cambridge 0580 Extended | E2.8 | Direct and inverse proportion in algebraic terms, with linear, square, square root, cube and cube root forms; the ∝ symbol |
| Edexcel 4MA1 Foundation | 1.7C, 1.7D | Use proportionality to find unknown quantities, e.g. s varies directly as t, complete a table |
| Edexcel 4MA1 Higher | 2.5A | Set up direct and inverse proportion with x, x^2, x^3, sqrt(x) and their reciprocals, and relate the algebra to graphs |
The key ideas
The symbol ∝ means 'is proportional to'. Replace '∝' with '= k ×' to make an equation: y ∝ x^2 becomes y = kx^2, and y ∝ 1/x becomes y = k/x. The constant k is the same for every pair of values, which is why one known pair is enough to find it.
Read the wording carefully. 'y is proportional to the square of x' is y = kx^2. 'y is inversely proportional to the square root of x' is y = k/sqrt(x). 'y varies as the cube of x' is y = kx^3.
Graphs: y = kx is a straight line through the origin; y = kx^2 is half a parabola through the origin for positive x; y = k/x is a curve that falls towards both axes without touching them.
Scaling questions can be done without k. If y ∝ x^2 and x is multiplied by 3, y is multiplied by 3^2 = 9. If y ∝ 1/x^2 and x doubles, y is divided by 4.
Worked example 1: direct proportion to a square
- y is directly proportional to x^2. When x = 2, y = 12. Find y when x = 4.5, and x when y = 27.
- Write the equation: y = kx^2.
- Find k: 12 = k × 4, so k = 3. The formula is y = 3x^2.
- When x = 4.5: y = 3 × 20.25 = 60.75.
- When y = 27: 27 = 3x^2, x^2 = 9, x = 3 (taking the positive root if x is a length or other positive quantity). Check: 3 × 9 = 27. Correct.
Worked example 2: inverse square
- The force F between two magnets is inversely proportional to the square of the distance d between them. F = 20 when d = 3.
- F = k/d^2, so 20 = k/9 and k = 180. The formula is F = 180/d^2.
- When d = 6: F = 180/36 = 5. Doubling the distance divides the force by 4, as expected (20 ÷ 4 = 5).
- When F = 45: 45 = 180/d^2, d^2 = 4, d = 2.
Worked example 3: square root and a percentage effect
- y ∝ sqrt(x), and y = 6 when x = 9. So y = k sqrt(x), 6 = 3k, k = 2. When x = 25, y = 2 × 5 = 10.
- Percentage effect: V ∝ r^3 and r increases by 10%. Then V is multiplied by 1.1^3 = 1.331, an increase of 33.1%.
- Check with numbers: if r goes from 10 to 11, r^3 goes from 1000 to 1331, an increase of 331, which is 33.1% of 1000. Correct.
Common mistakes that cost marks
- Using y = kx when the question says 'proportional to the square of x'.
- Writing y = x/k or y = k + x instead of y = kx or y = k/x.
- Squaring the given x value in the wrong place, or forgetting to square it, when finding k.
- Not writing the formula with the value of k: many questions award a mark for 'y = 3x^2' as a separate line.
- Forgetting that 'inversely proportional to the square root' means y = k/sqrt(x), with k on top.
- In scaling questions, multiplying by the scale factor instead of its square or cube.
Exam technique and how a tutor helps
These questions follow such a reliable pattern that they are some of the safest marks on an Extended or Higher paper. Always write three lines before answering any part: the equation with k, the substitution, and the formula with k found. If the question has a part (a) 'find a formula', that third line is the answer.
Look at a table of values to recognise the type: if multiplying x by 2 multiplies y by 4, it is a square; if it divides y by 2, it is inverse.
Students often lose marks here not through the algebra but by misreading which power applies. In one-to-one lessons a tutor works through the wording with the student, translating sentences into equations until the pattern is automatic, then links the equations to their graphs so that the shapes reinforce the meaning.
Self-check: can you do these?
- y ∝ x, and y = 14 when x = 4. Find y when x = 10. (Answer: 35)
- p is inversely proportional to q, and p = 8 when q = 3. Find p when q = 12. (Answer: 2)
- V ∝ r^3, and V = 54 when r = 3. Find V when r = 4. (Answer: 128)
- y is inversely proportional to sqrt(x), and y = 4 when x = 9. Find y when x = 16. (Answer: 3)
- y ∝ 1/x^2. What happens to y when x is doubled? (Answer: it is divided by 4)
Common questions
What does the symbol ∝ mean?
'Is proportional to'. y ∝ x means y = kx for some constant k. Cambridge 0580 states that knowledge of the ∝ symbol is required at Extended.
Is algebraic proportion on the Core or Foundation tier?
No. Equations like y = kx^2 are Cambridge Extended (E2.8) and Edexcel Higher (2.5A). Core and Foundation cover proportion through ratio, recipes, scales and simple direct proportion tables.
How do I know whether it is direct or inverse?
Direct: as one increases the other increases in step. Inverse: as one increases the other decreases. The question's wording ('directly' or 'inversely') settles it.
Do I need to find k every time?
For questions with numbers, yes: it is the most reliable method and usually earns a mark. For questions about the effect of a change, such as doubling x, you can reason with the power directly.
What does LiveTutor cost?
$15 for each one-to-one, 60-minute online lesson, at any level. Families choose 1 to 5 lessons a week on a monthly plan, and the first lesson is a free trial.
Sources
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