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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
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Teacher resources

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Student handouts

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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.

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Non-calculator 3 marks Easier

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.

Mark scheme — 3 marks available

  • 60 worker-days — M1
  • \(60 \div 8\) — M1
  • 7.5 — A1

Model answer

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

2. Exam question Non-calculator 3 marks Easier

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

Mark scheme — 3 marks available

  • \(y = kx\) — M1
  • \(k = 3\) — A1
  • 30 — A1

Model answer

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

3. Exam question Non-calculator 3 marks Easier

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

Mark scheme — 3 marks available

  • \(y = \dfrac{k}{x}\) — M1
  • \(k = 48\) — A1
  • 8 — A1

Model answer

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

4. Exam question Non-calculator 3 marks Easier

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.

Mark scheme — 3 marks available

  • Recognising inverse proportion — M1
  • \(k = 12\) — M1
  • \(y = \dfrac{12}{x}\) — A1

Model answer

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

5. Exam question Non-calculator 3 marks Easier

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

Mark scheme — 3 marks available

  • \(y = kx^2\) — M1
  • \(k = 5\) — A1
  • 45 — A1

Model answer

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

6. Exam question Non-calculator 3 marks Easier

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

Mark scheme — 3 marks available

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

Model answer

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

7. Multiple choice 1 mark Easier

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

  1. A It halves
  2. B It doubles Correct
  3. C It stays the same
  4. D It quadruples

Why: Direct proportion: \(y\) doubles too.

8. Multiple choice 1 mark Core

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

  1. A 12 days
  2. B 6 days
  3. C 3 days Correct
  4. D 24 days

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

9. Multiple choice 1 mark Core

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

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

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

10. Multiple choice 1 mark Core

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

  1. A 36
  2. B 9
  3. C 15
  4. D 4 Correct

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

11. Multiple choice 1 mark Core

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

  1. A A curve
  2. B A straight line through the origin Correct
  3. C A horizontal line
  4. D A pair of separate curves

Why: A straight line through the origin.

12. Multiple choice 1 mark Stretch

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

  1. A 30
  2. B 45
  3. C 50 Correct
  4. D 90

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