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Maths · Multiplicative reasoning

Ratio and proportion

Solve direct and inverse proportion problems, use the constant of proportionality, and write and use proportionality equations, including squares and roots at Higher tier.

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    4 pens cost £3.20. How much does 1 pen cost?

    Show answerHide answer

    £0.80

  2. 2

    If \(y = 3x\) and \(x = 5\), what is \(y\)?

    Show answerHide answer

    15

  3. 3

    Solve \(12 = \dfrac{k}{4}\).

    Show answerHide answer

    \(k = 48\)

  4. 4

    Work out \(48 \div 6\).

    Show answerHide answer

    8

  5. 5

    What does "directly proportional" mean?

    Show answerHide answer

    Both increase in the same ratio

Learning Objectives

  1. 1Solve direct proportion problems with \(y = kx\).
  2. 2Solve inverse proportion problems with \(y = \dfrac{k}{x}\).
  3. 3Recognise proportion from a table or graph.
  4. 4Use \(y \propto x^2\), \(y \propto x^3\) and \(y \propto \sqrt{x}\) (Higher).

Direct and Inverse Proportion

Direct proportion

  • As x increases, y increases in the same ratio.
  • \(y = kx\), so \(\dfrac{y}{x}\) is constant.
  • The graph is a straight line through the origin.

Inverse proportion

  • As x increases, y decreases in the opposite ratio.
  • \(y = \dfrac{k}{x}\), so \(xy\) is constant.
  • The graph is a curve that never touches the axes.

Direct Proportion

\(y\) is directly proportional to \(x\). When \(x = 7\), \(y = 21\). Find \(y\) when \(x = 10\).

Show the solutionHide the solution
  1. 1 Write \(y = kx\) \(21 = k \times 7\)
  2. 2 Find k \(k = 3\)
  3. 3 Substitute \(x = 10\) \(y = 3 \times 10\)

Answer\(y = 30\)

Inverse Proportion

\(y\) is inversely proportional to \(x\). When \(x = 4\), \(y = 12\). Find \(y\) when \(x = 6\).

Show the solutionHide the solution
  1. 1 Write \(y = \dfrac{k}{x}\) \(12 = \dfrac{k}{4}\)
  2. 2 Find k \(k = 48\)
  3. 3 Substitute \(x = 6\) \(y = \dfrac{48}{6}\)

Answer\(y = 8\)

A Worker Problem

5 workers take 12 days to build a wall. How long would 8 workers take, working at the same rate?

Show the solutionHide the solution
  1. 1 More workers means fewer days: inverse proportion Workers \(\times\) days is constant
  2. 2 Constant \(5 \times 12 = 60\)
  3. 3 Days for 8 workers \(60 \div 8\)

Answer7.5 days

Proportional to a Square

\(y\) is directly proportional to \(x^2\). When \(x = 2\), \(y = 20\). Find \(y\) when \(x = 3\).

Show the solutionHide the solution
  1. 1 Write \(y = kx^2\) \(20 = k \times 4\)
  2. 2 Find k \(k = 5\)
  3. 3 Substitute \(x = 3\) \(y = 5 \times 9\)

Answer\(y = 45\)

Proportional to a Square Root

\(y\) is directly proportional to \(\sqrt{x}\). When \(x = 4\), \(y = 6\). Find \(y\) when \(x = 25\).

Show the solutionHide the solution
  1. 1 Write \(y = k\sqrt{x}\) \(6 = k \times 2\)
  2. 2 Find k \(k = 3\)
  3. 3 Substitute \(x = 25\) \(y = 3 \times 5\)

Answer\(y = 15\)

Types of Proportion

Learn the equation for each phrase.

  • \(y\) is proportional to \(x\)

    Equation: \(y = kx\)

  • \(y\) is inversely proportional to \(x\)

    Equation: \(y = \dfrac{k}{x}\)

  • \(y\) is proportional to \(x^2\)

    Equation: \(y = kx^2\)

  • \(y\) is proportional to \(x^3\)

    Equation: \(y = kx^3\)

  • \(y\) is proportional to \(\sqrt{x}\)

    Equation: \(y = k\sqrt{x}\)

  • \(y\) is inversely proportional to \(x^2\)

    Equation: \(y = \dfrac{k}{x^2}\)

Which Type?

For each situation decide whether it is direct or inverse proportion, and write an equation with a constant \(k\). (a) The cost of buying \(n\) identical books (b) the time \(t\) to fill a pool with \(p\) identical pumps (c) the area \(A\) of a circle with radius \(r\).

1. Say what happens when one doubles.

2. Write the equation.

A good answer shows: (a) Direct: \(C = kn\). (b) Inverse: \(t = \dfrac{k}{p}\). (c) \(A = kr^2\) (with \(k = \pi\)).

Can I...?

  1. 1Solve a direct proportion problem.
  2. 2Solve an inverse proportion problem.
  3. 3Find the constant \(k\).
  4. 4Write \(y = kx\) and \(y = \dfrac{k}{x}\).
  5. 5Recognise proportion from a graph.
  6. 6Use \(y = kx^2\) (Higher).
  7. 7Use \(y = k\sqrt{x}\) (Higher).
  8. 8Explain the type of proportion.

Summary & Exam Focus

  • Direct: \(y = kx\). Inverse: \(y = \dfrac{k}{x}\).
  • Find \(k\) from the first pair of values.
  • Higher: \(y = kx^2\), \(y = kx^3\), \(y = k\sqrt{x}\).
  • Check your answer makes sense.

Exam focus

\(y\) is directly proportional to \(x^2\). When \(x = 2\), \(y = 20\). Find the value of \(y\) when \(x = 3\). (3 marks) (3 marks)

Write the equation with \(k\) first, use the given pair to find \(k\), then substitute the new value.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Direct proportion
Two quantities that increase together in the same ratio.
Inverse proportion
As one quantity increases the other decreases in the opposite ratio.
Constant of proportionality
The number \(k\) in a proportion equation.
Proportional to
Written \(y \propto x\).
Unitary method
Finding the value of one unit first.
Rate
How much of one quantity per unit of another.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Non-calculator 3 marks

    5 workers can build a wall in 12 days. All the workers work at the same rate. Work out how many days it would take 8 workers to build the wall.

    Show answerHide answer

    Model answer

    \(5 \times 12 = 60\) worker-days. \(60 \div 8 = 7.5\) days.

    Mark scheme

    • 60 worker-days — M1
    • \(60 \div 8\) — M1
    • 7.5 — A1
  2. Question 2 Non-calculator 3 marks

    \(y\) is directly proportional to \(x\). When \(x = 7\), \(y = 21\). Find the value of \(y\) when \(x = 10\).

    Show answerHide answer

    Model answer

    \(y = kx\), so \(21 = 7k\) and \(k = 3\). When \(x = 10\), \(y = 30\).

    Mark scheme

    • \(y = kx\) — M1
    • \(k = 3\) — A1
    • 30 — A1
  3. Question 3 Non-calculator 3 marks

    \(y\) is inversely proportional to \(x\). When \(x = 4\), \(y = 12\). Find the value of \(y\) when \(x = 6\).

    Show answerHide answer

    Model answer

    \(y = \dfrac{k}{x}\), so \(12 = \dfrac{k}{4}\) and \(k = 48\). When \(x = 6\), \(y = 8\).

    Mark scheme

    • \(y = \dfrac{k}{x}\) — M1
    • \(k = 48\) — A1
    • 8 — A1
  4. Question 4 Non-calculator 3 marks

    The graph shows the relationship between \(x\) and \(y\). Write down an equation connecting \(x\) and \(y\).

    A falling curve through (1, 12), (2, 6), (3, 4), (4, 3), (6, 2) and (12, 1), showing y equal to 12 divided by x.
    Show answerHide answer

    Model answer

    The points show that \(xy = 12\) each time, so \(y = \dfrac{12}{x}\) (inverse proportion).

    Mark scheme

    • Recognising inverse proportion — M1
    • \(k = 12\) — M1
    • \(y = \dfrac{12}{x}\) — A1
  5. Question 5 Non-calculator 3 marks

    \(y\) is directly proportional to \(x^2\). When \(x = 2\), \(y = 20\). Find the value of \(y\) when \(x = 3\).

    Show answerHide answer

    Model answer

    \(y = kx^2\), so \(20 = 4k\) and \(k = 5\). When \(x = 3\), \(y = 5 \times 9 = 45\).

    Mark scheme

    • \(y = kx^2\) — M1
    • \(k = 5\) — A1
    • 45 — A1
  6. Question 6 Non-calculator 3 marks

    \(y\) is directly proportional to \(\sqrt{x}\). When \(x = 4\), \(y = 6\). Find the value of \(y\) when \(x = 25\).

    Show answerHide answer

    Model answer

    \(y = k\sqrt{x}\), so \(6 = 2k\) and \(k = 3\). When \(x = 25\), \(y = 3 \times 5 = 15\).

    Mark scheme

    • \(y = k\sqrt{x}\) — M1
    • \(k = 3\) — A1
    • 15 — A1

Quick check

  1. \(y = 4x\). What happens to \(y\) when \(x\) doubles?

    1. AIt halves
    2. BIt doubles
    3. CIt stays the same
    4. DIt quadruples
    Show answerHide answer

    B: It doubles

    Direct proportion: \(y\) doubles too.

  2. 4 people take 6 days to paint a fence. How long do 8 people take?

    1. A12 days
    2. B6 days
    3. C3 days
    4. D24 days
    Show answerHide answer

    C: 3 days

    Inverse proportion: \(4 \times 6 = 24\), and \(24 \div 8 = 3\) days.

  3. Which equation shows \(y\) inversely proportional to \(x\)?

    1. A\(y = \dfrac{k}{x}\)
    2. B\(y = kx\)
    3. C\(y = kx^2\)
    4. D\(y = k + x\)
    Show answerHide answer

    A: \(y = \dfrac{k}{x}\)

    \(y = \dfrac{k}{x}\).

  4. \(y = kx\) with \(y = 12\) when \(x = 3\). What is \(k\)?

    1. A36
    2. B9
    3. C15
    4. D4
    Show answerHide answer

    D: 4

    \(k = 12 \div 3 = 4\).

  5. The graph of \(y = kx\) is...

    1. AA curve
    2. BA straight line through the origin
    3. CA horizontal line
    4. DA pair of separate curves
    Show answerHide answer

    B: A straight line through the origin

    A straight line through the origin.

  6. \(y \propto x^2\) and \(y = 18\) when \(x = 3\). What is \(y\) when \(x = 5\)?

    1. A30
    2. B45
    3. C50
    4. D90
    Show answerHide answer

    C: 50

    \(k = 2\), so \(y = 2 \times 25 = 50\).

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