← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 2 Algebra and graphs · Chapter 16 · pp. 166–171

Proportion

Two quantities are in proportion when one is always a fixed multiple of the other (or of its square, its root or its reciprocal). Write the link as an equation with a constant k, find k from one known pair, and you can predict any other value. The whole chapter is Extended content.

Learning objectives

  1. Recognise direct proportion, y ∝ x, and write it as y = kx.
  2. Know that k is the constant of proportionality and the gradient of the graph.
  3. Work with proportion to a power or root: y ∝ x², y ∝ x³, y ∝ √x.
  4. Work with inverse proportion: y ∝ , y ∝ .
  5. Find k from one pair of values, then find unknown values.
  6. Turn word problems into proportion equations.

Book exercises: 16.1, 16.2 and 16.3, and Student assessments 1 and 2 (pp. 166–171).

Direct proportion Extended

🔑 Key ideas

  • y ∝ x means “y is directly proportional to x”.
  • It becomes an equation with a constant: y = kx.
  • k is the constant of proportionality (or constant of variation).
  • Divide any y by its x and you always get k: = k.
  • The graph of y against x is a straight line through the origin with gradient k.
x0123510k = y ÷ x
y = 2x024610202
y = 3x036915303
y = 2.5x02.557.512.5252.5
0246810 102030 x y y = 3x (k = 3) y = 2.5x (k = 2.5) y = 2x (k = 2) run 2 rise 4 gradient = 4 ÷ 2 = 2 = k
All three tables give straight lines through the origin. The steeper the line, the bigger k: k is the gradient.

📘 Example 1 (book) · y ∝ x. If y = 7 when x = 2, find y when x = 5

  1. Write the proportion as an equation: y = kx.
  2. Substitute the known pair: 7 = k × 2.
  3. Solve for k: k = 7 ÷ 2 = 3.5, so y = 3.5x.
  4. Use x = 5: y = 3.5 × 5 = 17.5.
  5. Check: 17.5 ÷ 5 = 3.5 = 7 ÷ 2. ✔

📘 Example 2 · The cost C of fuel is proportional to the litres L bought. 20 litres cost $30. How many litres can you buy for $60?

  1. C ∝ L, so C = kL.
  2. Substitute: 30 = k × 20, so k = 30 ÷ 20 = 1.5 (the price per litre).
  3. Use C = 60: 60 = 1.5L.
  4. Divide: L = 60 ÷ 1.5 = 40 litres.
  5. Check: double the money buys double the fuel (20 → 40). ✔

✏️ Try it

  1. y ∝ x and y = 12 when x = 4. Find k.
  2. y ∝ x and y = 9 when x = 6. Find y when x = 10.
  3. y ∝ x and y = 20 when x = 8. Find x when y = 35.
Answers

1. k = 12 ÷ 4 = 3 · 2. k = 1.5, y = 15 · 3. k = 2.5, x = 35 ÷ 2.5 = 14

Next: proportion to powers and roots →

Proportion to powers and roots Extended

🔑 Key ideas

  • y can be proportional to a power of x: y ∝ xⁿ (n > 0) gives y = kxⁿ.
  • Common ones: y ∝ x² (y = kx²), y ∝ x³ (y = kx³), y ∝ √x (y = k√x, since √x = x1/2).
  • The graph of y against x is now a curve.
  • Plot y against the power instead (x², x³ or √x) and it becomes a straight line through the origin.
x12345
y = 2x²28183250
y = ½x³½413½3262½
y = √x1√2√32√5
y against x: a curve 12345 1020304050 x y = 2x² y against x²: a straight line 510152025 1020304050 x² gradient = k = 2 Same five points (1, 2), (2, 8), (3, 18), (4, 32), (5, 50): only the x-axis changes.
y = 2x² curves when plotted against x, but is a straight line through the origin when plotted against x². That line proves y ∝ x².

What happens to y when x doubles?

ProportionEquationx doubles, so y is…
y ∝ xy = kx× 2
y ∝ x²y = kx²× 4
y ∝ x³y = kx³× 8
y ∝ √xy = k√x× √2 (about 1.41)
y ∝ 1/xy = k/x÷ 2
y ∝ 1/x²y = k/x²÷ 4

📘 Example 3 · y ∝ x² and y = 20 when x = 2. Find y when x = 6, and x when y = 125

  1. Write the equation: y = kx².
  2. Substitute: 20 = k × 2² = 4k.
  3. Solve: k = 20 ÷ 4 = 5, so y = 5x².
  4. x = 6: y = 5 × 6² = 5 × 36 = 180.
  5. y = 125: 125 = 5x², so x² = 25.
  6. Square root: x = 5 (or −5 if x can be negative).

📘 Example 4 · y ∝ √x and y = 6 when x = 9. Find y when x = 49

  1. Write the equation: y = k√x.
  2. Substitute: 6 = k × √9 = 3k.
  3. Solve: k = 2, so y = 2√x.
  4. x = 49: y = 2 × √49 = 2 × 7 = 14.

📘 Example 5 · y ∝ x³ and y = 16 when x = 2. Find y when x = 3

  1. y = kx³, so 16 = k × 2³ = 8k.
  2. k = 16 ÷ 8 = 2, so y = 2x³.
  3. x = 3: y = 2 × 27 = 54.

✏️ Try it

  1. y ∝ x² and y = 27 when x = 3. Find y when x = 5.
  2. y ∝ √x and y = 15 when x = 25. Find x when y = 12.
  3. y ∝ x³. If x is multiplied by 3, what is y multiplied by?
Answers

1. k = 3, y = 75 · 2. k = 3, √x = 4, x = 16 · 3. 3³ = 27

Next: inverse proportion →

Inverse proportion Extended

🔑 Key ideas

  • y is inversely proportional to x: y ∝ , so y = .
  • As x goes up, y goes down. Double x and y halves.
  • Multiply any pair together and you always get k: xy = k.
  • The same idea works with powers: y ∝ gives y = , y ∝ gives y = .
  • The graph of y against is a straight line through the origin.
24681012 24681012 x y y = 12/x x × y 2 × 6 = 123 × 4 = 124 × 3 = 126 × 2 = 12 always k = 12
For y = 12/x every point makes a rectangle of area 12 under the curve (two are shaded): x × y never changes, and that constant is k.

📘 Example 6 (book) · y ∝ . If y = 5 when x = 3, find y when x = 30

  1. Write the equation: y = .
  2. Substitute: 5 = .
  3. Multiply both sides by 3: k = 15, so y = .
  4. x = 30: y = = 0.5.
  5. Check: 3 × 5 = 15 and 30 × 0.5 = 15. ✔

📘 Example 7 · y ∝ and y = 5 when x = 2. Find y when x = 4, and x when y = 0.8

  1. Write the equation: y = .
  2. Substitute: 5 = , so k = 20.
  3. x = 4: y = = 1.25.
  4. y = 0.8: 0.8 = , so x² = 20 ÷ 0.8 = 25.
  5. Square root: x = 5 (or −5).

✏️ Try it

  1. y ∝ and y = 4 when x = 6. Find y when x = 8.
  2. y ∝ and y = 9 when x = 2. Find y when x = 3.
  3. y ∝ and y = 3 when x = 16. Find y when x = 9.
Answers

1. k = 24, y = 3 · 2. k = 36, y = 36 ÷ 9 = 4 · 3. k = 12, y = 12 ÷ 3 = 4

Next: the four-step method →

The four-step method Extended

Every proportion question, from a bare “y ∝ x²” to a cliff and a falling stone, is solved the same way.

1. Write ∝y ∝ x² 2. Add ky = kx² 3. Find k20 = k × 4, k = 5 4. Predicty = 5x² Step 3 needs one known pair of values. Step 4 works forwards (find y) or backwards (find x).
Write the proportion, turn it into an equation with k, use the known pair to find k, then use the full equation.

📘 Example 8 · m ∝ and m = 2 when r = 3. Find m when r = 6

  1. Write ∝: m ∝ .
  2. Add k: m = .
  3. Find k: 2 = , so k = 54.
  4. Predict: m = = = 0.25.
  5. Check: r doubled, so m should be divided by 2³ = 8. 2 ÷ 8 = 0.25. ✔

🧪 Try it: variation solver

Known pair:

    ✏️ Try it

    1. p ∝ q² and p = 12 when q = 2. Find p when q = 3.
    2. y ∝ and y = 10 when x = 1.5. Find x when y = 3.
    3. a ∝ √b and a = 8 when b = 16. Find b when a = 14.
    Answers

    1. k = 3, p = 27 · 2. k = 15, x = 5 · 3. k = 2, √b = 7, b = 49

    Next: word problems →

    Word problems Extended

    The hard part of a word problem is the first step: turning the sentence into symbols. After that it is the four-step method.

    The sentence Proportion Equation h is proportional to the square of t h ∝ t² h = kt² v is proportional to the square root of e v ∝ √e v = k√e P is inversely proportional to V P ∝ 1/V P = k/V F is inversely proportional to the square of d F ∝ 1/d² F = k/d²
    Pick out the two letters, the word “inversely” (or not), and the power: “square” is ², “cube” is ³, “square root” is √.

    📘 Example 9 · A car's stopping distance d m is proportional to the square of its speed v km/h. At 40 km/h, d = 20 m

    (a) Find d at 80 km/h. (b) Find v when d = 45 m.

    1. Write ∝ and add k: d ∝ v², so d = kv².
    2. Find k: 20 = k × 40² = 1600k, so k = 20 ÷ 1600 = 0.0125.
    3. (a) v = 80: d = 0.0125 × 80² = 0.0125 × 6400 = 80 m.
    4. Check (a): speed doubled, so distance × 4: 20 × 4 = 80. ✔
    5. (b) 45 = 0.0125v², so v² = 45 ÷ 0.0125 = 3600.
    6. Square root (speed is positive): v = 60 km/h.

    📘 Example 10 · The time T days to build a wall is inversely proportional to the number of workers n. 6 workers take 10 days. How long do 4 workers take?

    1. T ∝ , so T = .
    2. Find k: 10 = , so k = 60 (60 worker-days of work).
    3. n = 4: T = = 15 days.
    4. Check: fewer workers should take longer, and 15 > 10. ✔

    📘 Example 11 · Light intensity I is inversely proportional to the square of the distance d from the lamp. I = 36 when d = 2. Find I when d = 3

    1. I = .
    2. Find k: 36 = , so k = 144.
    3. d = 3: I = = 16.

    ✏️ Try it

    1. The area A of a picture is proportional to the square of its width w. A = 48 cm² when w = 4 cm. Find A when w = 10 cm.
    2. The time t to drive a fixed route is inversely proportional to the speed s. t = 3 hours at 60 km/h. Find t at 90 km/h.
    3. Write as an equation: “the period T is proportional to the square root of the length l”.
    Answers

    1. k = 3, A = 300 cm² · 2. k = 180, t = 2 hours · 3. T = k√l

    Next: summary and quiz →

    Chapter summary

    • y ∝ x means y = kx: a straight line through the origin with gradient k.
    • y ∝ xⁿ means y = kxⁿ (x², x³, √x). Plot y against xⁿ to get a straight line.
    • y ∝ 1/x means y = k/x: as x goes up, y goes down, and xy = k.
    • Four steps: write ∝, add k, find k from a known pair, then predict.
    • Doubling x: y × 2 (x), × 4 (x²), × 8 (x³), ÷ 2 (1/x), ÷ 4 (1/x²).
    • When you square-root to find x, there may be two answers (±). Use the positive one for lengths, times and speeds.

    📄 Practice questions

    The book's Exercises 16.1 to 16.3 and Student assessments 1 and 2 are in the printable assignment for this chapter.

    Mini-quiz

    1. y is directly proportional to x. Which equation shows this?
    2. The graph of y against x for y ∝ x is always…
    3. y ∝ x and y = 12 when x = 3. What is k?
    4. y ∝ x and y = 10 when x = 4. Find y when x = 6.
    5. y ∝ x². If x is doubled, what happens to y?
    6. y ∝ x² and y = 45 when x = 3. Find y when x = 4.
    7. y is inversely proportional to x. Which equation shows this?
    8. y ∝ 1/x and y = 6 when x = 4. What is k?
    9. y ∝ 1/x and y = 8 when x = 3. Find y when x = 12.
    10. y ∝ 1/x². If x is doubled, what happens to y?
    11. y ∝ √x and y = 6 when x = 9. Find y when x = 25.
    12. Which graph is a straight line through the origin when y ∝ x²?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 16 (Proportion). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.