IGCSE Maths · Algebra and Graphs · Extended only

Proportion: Direct and Inverse Relationships Extended only

This entire note is Extended-only. It’s the algebraic version of the proportion work from the Number unit’s Ratio, Proportion and Rates note — the same idea of “one quantity scaling with another,” now written formally with the symbol ∝ and turned into an equation you can use for any values, not just the ones you’re directly given.

Why this matters “Directly proportional” and “inversely proportional” describe an enormous range of real relationships — the cost of a cake scaling with the square of its diameter, the time a job takes shrinking as more workers join, the brightness of a light fading with the square of your distance from it. Writing these as y = kxⁿ turns a vague “connected somehow” into a precise, usable formula.

1. Direct Proportion

“y is directly proportional to x” is written y ∝ x, and means y = kx for some constant k, called the constant of proportionality. As x increases, y increases at the same steady rate.

y ∝ x   ⟺   y = kx
Worked example y is directly proportional to x. When x = 3, y = 15. Find the formula connecting y and x, then find y when x = 8.
y = kx → 15 = k(3) → k = 5
Formula: y = 5x
When x = 8: y = 5(8) = 40
Try it — Direct Proportion
  1. y is directly proportional to x. When x = 4, y = 20. Find the formula, then find y when x = 9.
  2. p is directly proportional to q. When q = 6, p = 21. Find p when q = 10.
Show answers
  1. y = kx → 20 = 4k → k = 5. y = 5x. When x = 9: y = 45
  2. p = kq → 21 = 6k → k = 3.5. p = 3.5q. When q = 10: p = 35

2. Direct Proportion with Powers and Roots

The same idea extends to squares, cubes, and roots — only the power on x changes.

StatementEquation
y ∝ x²y = kx²
y ∝ x³y = kx³
y ∝ √xy = k√x
y ∝ ³√xy = k³√x
Worked example — square proportion y is directly proportional to x². When x = 4, y = 48. Find y when x = 5.
y = kx² → 48 = k(16) → k = 3
Formula: y = 3x². When x = 5: y = 3(25) = 75
Worked example — square root proportion y is directly proportional to √x. When x = 9, y = 12. Find y when x = 25.
y = k√x → 12 = k√9 = 3k → k = 4
Formula: y = 4√x. When x = 25: y = 4(5) = 20
Worked example — cube proportion, finding x y is directly proportional to x³. When x = 3, y = 54. Find x when y = 128.
y = kx³ → 54 = k(27) → k = 2
Formula: y = 2x³. When y = 128: 128 = 2x³ → x³ = 64 → x = 4
Try it — Direct Proportion with Powers
  1. y is directly proportional to x². When x = 3, y = 27. Find y when x = 6.
  2. y is directly proportional to x³. When x = 2, y = 40. Find y when x = 5.
  3. y is directly proportional to √x. When x = 16, y = 20. Find x when y = 30.
Show answers
  1. 27 = 9k → k = 3. y = 3x². When x = 6: y = 108
  2. 40 = 8k → k = 5. y = 5x³. When x = 5: y = 625
  3. 20 = k(4) → k = 5. y = 5√x. When y = 30: 30 = 5√x → √x = 6 → x = 36

3. Inverse Proportion

“y is inversely proportional to x” is written y ∝ 1/x, and means y = k/x. As x increases, y decreases — they move in opposite directions.

y ∝ 1/x   ⟺   y = k/x
Worked example y is inversely proportional to x. When x = 4, y = 6. Find y when x = 8.
y = k/x → 6 = k/4 → k = 24
Formula: y = 24/x. When x = 8: y = 24/8 = 3
Try it — Inverse Proportion
  1. y is inversely proportional to x. When x = 5, y = 8. Find y when x = 10.
  2. p is inversely proportional to q. When q = 3, p = 20. Find q when p = 4.
Show answers
  1. 8 = k/5 → k = 40. y = 40/x. When x = 10: y = 4
  2. 20 = k/3 → k = 60. p = 60/q. When p = 4: 4 = 60/q → q = 15

4. Inverse Proportion with Powers and Roots

StatementEquation
y ∝ 1/x²y = k/x²
y ∝ 1/x³y = k/x³
y ∝ 1/√xy = k/√x
Worked example y is inversely proportional to x². When x = 2, y = 5. Find y when x = 4.
y = k/x² → 5 = k/4 → k = 20
Formula: y = 20/x². When x = 4: y = 20/16 = 1.25
Worked example — inverse square root, finding x y is inversely proportional to √x. When x = 4, y = 10. Find x when y = 4.
y = k/√x → 10 = k/√4 = k/2 → k = 20
Formula: y = 20/√x. When y = 4: 4 = 20/√x → √x = 5 → x = 25
Try it — Inverse Proportion with Powers
  1. y is inversely proportional to x². When x = 3, y = 4. Find y when x = 6.
  2. y is inversely proportional to x³. When x = 2, y = 5. Find y when x = 1.
  3. y is inversely proportional to √x. When x = 9, y = 6. Find x when y = 2.
Show answers
  1. 4 = k/9 → k = 36. y = 36/x². When x = 6: y = 1
  2. 5 = k/8 → k = 40. y = 40/x³. When x = 1: y = 40
  3. 6 = k/3 → k = 18. y = 18/√x. When y = 2: 2 = 18/√x → √x = 9 → x = 81

5. Solving Real-World Proportion Problems

The method is always the same three steps: (1) write the proportion statement as an equation with k, (2) use the given values to find k, (3) use the completed formula to answer the actual question.

Worked example — direct square proportion The cost C of a circular cake is directly proportional to the square of its diameter d. A cake with diameter 20 cm costs $32. Find the cost of a cake with diameter 25 cm.
C = kd² → 32 = k(400) → k = 0.08
Formula: C = 0.08d². When d = 25: C = 0.08(625) = $50
Worked example — inverse proportion The time T (in hours) taken to complete a job is inversely proportional to the number of workers n. 6 workers take 10 hours. How long would 15 workers take?
T = k/n → 10 = k/6 → k = 60
Formula: T = 60/n. When n = 15: T = 60/15 = 4 hours
Try it — Real-World Problems
  1. The energy E stored in a spring is directly proportional to the square of its extension x. When x = 3 cm, E = 18 J. Find E when x = 5 cm.
  2. The pressure P of a gas is inversely proportional to its volume V. When V = 4 m³, P = 150 kPa. Find P when V = 6 m³.
  3. The distance d a stone falls is directly proportional to the square of the time t it has been falling. After 2 seconds it has fallen 20 m. How far has it fallen after 5 seconds?
Show answers
  1. E=kx² → 18=9k → k=2. E=2x². When x=5: E=50 J
  2. P=k/V → 150=k/4 → k=600. P=600/V. When V=6: P=100 kPa
  3. d=kt² → 20=4k → k=5. d=5t². When t=5: d=125 m
Common mistakes to watch for
  • Writing y = kx when the question actually says “y is proportional to x²” (or another power) — always match the equation to the exact power stated.
  • Mixing up direct and inverse — direct proportion means both quantities increase together; inverse means one increases as the other decreases.
  • Forgetting to find k first — you cannot answer the second part of a proportion question without it.
  • For inverse proportion, writing y = kx instead of y = k/x.
  • Rounding k too early — keep it exact (as a fraction or exact decimal) until the final answer.

6. Quick-Fire Challenge Round

True or False?
  1. If y ∝ x, then doubling x doubles y.
  2. If y ∝ x², then doubling x doubles y.
  3. If y ∝ 1/x, then doubling x halves y.
  4. The constant of proportionality can be negative.
Show answers
  1. True.
  2. False — doubling x means (2x)² = 4x², so y becomes 4 times bigger, not 2 times.
  3. True.
  4. True — nothing in the definition requires k to be positive, though most real-world contexts (cost, time, distance) do use positive values.
Spot the Error
  1. “y is proportional to x². When x=2, y=12, so k=6 and y=6x.”
  2. “y is inversely proportional to x. When x=5, y=4, so k=0.8 and y=0.8x.”
  3. “If y ∝ x³ and x triples, y also triples.”
Show answers
  1. Since y ∝ x², the equation is y = kx², not y = kx — so k = 12/4 = 3, giving y = 3x², not y = 6x.
  2. For inverse proportion, y = k/x, not y = kx: k = xy = 5 × 4 = 20, so y = 20/x, not y = 0.8x.
  3. If x triples, x³ becomes 27 times bigger (3³ = 27), so y becomes 27 times bigger, not just 3 times.

7. Mixed Practice — Bringing It All Together

  1. y is directly proportional to x. When x = 7, y = 21. Find y when x = 12.
  2. y is directly proportional to x². When x = 5, y = 75. Find y when x = 2.
  3. y is inversely proportional to x. When x = 3, y = 20. Find y when x = 5.
  4. y is inversely proportional to x². When x = 2, y = 9. Find x when y = 1.
  5. The cost of paint needed is directly proportional to the surface area to be covered. 30 m² needs $45 of paint. Find the cost for 50 m².
Show answers
  1. 21=7k → k=3. y=3x. When x=12: y=36
  2. 75=25k → k=3. y=3x². When x=2: y=12
  3. 20=k/3 → k=60. y=60/x. When x=5: y=12
  4. 9=k/4 → k=36. y=36/x². When y=1: x²=36 → x=6
  5. C=kA → 45=30k → k=1.5. C=1.5A. When A=50: C=$75

Practice at Home

A large set again — take it steadily.

Part A — Direct proportion (linear)
  1. y is directly proportional to x. When x = 6, y = 27. Find y when x = 10.
  2. a is directly proportional to b. When b = 8, a = 12. Find b when a = 30.
Part B — Direct proportion (powers and roots)
  1. y is directly proportional to x². When x = 4, y = 32. Find y when x = 7.
  2. y is directly proportional to x³. When x = 2, y = 16. Find y when x = 4.
  3. y is directly proportional to √x. When x = 4, y = 6. Find x when y = 15.
Part C — Inverse proportion
  1. y is inversely proportional to x. When x = 4, y = 15. Find y when x = 6.
  2. y is inversely proportional to x². When x = 3, y = 8. Find y when x = 2.
  3. y is inversely proportional to √x. When x = 16, y = 3. Find x when y = 12.
Part D — Real-world problems
  1. The area A of a circle’s shadow is directly proportional to the square of the object’s height h above the ground. When h = 2 m, A = 8 m². Find A when h = 5 m.
  2. The number of days d needed to build a wall is inversely proportional to the number of workers w. 4 workers take 15 days. How many days would 6 workers take?
  3. The force F of attraction between two magnets is inversely proportional to the square of the distance x between them. When x = 2 cm, F = 40 N. Find F when x = 4 cm.
Part E — Applied / challenge
  1. y is directly proportional to x². When x = 3, y = 45. Find the value of x when y = 180.
  2. Two variables p and q satisfy p ∝ 1/q². If q is tripled, by what factor does p change? Explain your reasoning without picking specific numbers.
  3. The volume V of a sphere is directly proportional to the cube of its radius r. A sphere of radius 3 cm has volume 113.1 cm³ (to 1 d.p.). Use this to find the constant of proportionality, then predict the volume of a sphere with radius 6 cm — and explain why your answer makes sense compared to doubling the radius.

Key Vocabulary Recap

∝ — the symbol for “is proportional to”
Constant of proportionality (k) — the fixed number connecting the two quantities
Direct proportion — both quantities increase or decrease together
Inverse proportion — one quantity increases as the other decreases

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section E2.8 (Extended subject content) — Proportion, including linear, square, square root, cube and cube root direct and inverse proportion, and use of the ∝ symbol. There is no Core equivalent — the corresponding Core section (C2.8) is explicitly marked “Extended content only” in the syllabus.

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