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Exam questions · Maths · Ratio and Proportion

Direct and Inverse Proportion

  • 6 exam questions
  • 19 marks
  • 9 quick checks
  1. 1 Work out [3 marks]

    Here are the ingredients for 8 pancakes: 200 g of flour, 2 eggs and 300 ml of milk. Work out the amounts needed for 20 pancakes.

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    Model answer

    The scale factor is \(20 \div 8 = 2.5\). Flour: \(200 \times 2.5 = 500\) g. Eggs: \(2 \times 2.5 = 5\). Milk: \(300 \times 2.5 = 750\) ml.

    Mark scheme

    • Scale factor \(2.5\) or the amounts for 4 pancakes found (100 g, 1 egg, 150 ml) — M1
    • Two of the three amounts correct — A1
    • 500 g, 5 eggs and 750 ml — A1
  2. 2 Work out [3 marks]

    5 identical tins of beans cost \(\pounds 3.40\) altogether. Work out the cost of 8 of these tins.

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    Model answer

    One tin costs \(3.40 \div 5 = \pounds 0.68\), so 8 tins cost \(8 \times 0.68 = \pounds 5.44\).

    Mark scheme

    • \(3.40 \div 5 = 0.68\) — M1
    • \(8 \times 0.68\) or \(8 \times 68\) — M1
    • \(\pounds 5.44\) — A1
  3. 3 Work out [3 marks]

    Shampoo is sold in two sizes. Large bottle (750 ml) costs \(\pounds 4.50\) Small bottle (400 ml) costs \(\pounds 2.60\) Which bottle is the better value for money? You must show how you get your answer.

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    Model answer

    Large bottle: \(450 \div 750 = 0.6\) pence per ml. Small bottle: \(260 \div 400 = 0.65\) pence per ml. The large bottle costs less per millilitre, so it is better value.

    Mark scheme

    • A method to compare, such as the cost of 1 ml or the amount for 1 penny — M1
    • \(0.6\) and \(0.65\) pence per ml, or equivalent figures — A1
    • Large bottle, with a correct comparison — C1
  4. 4 Work out [3 marks]

    The exchange rate is \(\pounds 1 = \$1.25\). A jacket costs \(\$60\) in New York and \(\pounds 50\) in London. In which city is the jacket cheaper? You must show your working.

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    Model answer

    Change the New York price to pounds: \(60 \div 1.25 = 48\), so it costs \(\pounds 48\). Alternatively \(\pounds 50 = 50 \times 1.25 = \$62.50\). The jacket is cheaper in New York.

    Mark scheme

    • \(60 \div 1.25\) or \(50 \times 1.25\) — M1
    • \(\pounds 48\) or \(\$62.50\) — A1
    • New York, with a correct comparison — C1
  5. 5 Work out [3 marks]

    It takes 4 workers 6 days to paint a house. All the workers paint at the same rate. Work out how long it would take 3 workers to paint the house.

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    Model answer

    The job needs \(4 \times 6 = 24\) worker-days. With 3 workers it takes \(24 \div 3 = 8\) days.

    Mark scheme

    • \(4 \times 6 = 24\) — M1
    • \(24 \div 3\) — M1
    • 8 days — A1
  6. 6 Work out [4 marks]

    \(y\) is inversely proportional to \(x\). When \(x = 5\), \(y = 8\). (a) Find a formula for \(y\) in terms of \(x\). (2 marks) (b) Work out the value of \(y\) when \(x = 10\). (2 marks)

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    Model answer

    (a) \(y = \dfrac{k}{x}\), so \(8 = \dfrac{k}{5}\) and \(k = 40\). The formula is \(y = \dfrac{40}{x}\). (b) \(y = \dfrac{40}{10} = 4\).

    Mark scheme

    • (a) \(y = \dfrac{k}{x}\) with \(k = 40\) found — M1
    • (a) \(y = \dfrac{40}{x}\) — A1
    • (b) \(\dfrac{40}{10}\) — M1
    • (b) 4 — A1

Quick check

  1. 1

    5 pens cost \(\pounds 3.50\). How much do 8 pens cost?

    1. A\(\pounds 4.50\)
    2. B\(\pounds 6.30\)
    3. C\(\pounds 5.60\)
    4. D\(\pounds 2.80\)
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    C: \(\pounds 5.60\)

    One pen costs \(3.50 \div 5 = 0.70\). Then 8 pens cost \(8 \times 0.70 = 5.60\).

  2. 2

    A recipe for 4 people uses 300 g of flour. How much flour is needed for 10 people?

    1. A600 g
    2. B750 g
    3. C1200 g
    4. D450 g
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    B: 750 g

    For one person, \(300 \div 4 = 75\) g. For 10 people, \(75 \times 10 = 750\) g.

  3. 3

    Which of these shows that \(y\) is directly proportional to \(x\)?

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

    In direct proportion \(y\) is a constant multiple of \(x\), so \(y = 3x\), which makes a straight line through the origin.

  4. 4

    6 workers take 10 days to build a wall. How long would 15 workers take, working at the same rate?

    1. A25 days
    2. B9 days
    3. C1.5 days
    4. D4 days
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    D: 4 days

    The job takes \(6 \times 10 = 60\) worker-days. With 15 workers it takes \(60 \div 15 = 4\) days.

  5. 5

    Which pack of cereal is the best value?

    1. A500 g for \(\pounds 2.00\)
    2. B1 kg for \(\pounds 3.95\)
    3. C750 g for \(\pounds 2.85\)
    4. D2 kg for \(\pounds 7.80\)
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    C: 750 g for \(\pounds 2.85\)

    The cost per 100 g is 40p, 38p, 39.5p and 39p. The 750 g pack is the cheapest per 100 g.

  6. 6

    \(\pounds 1 = \euro 1.20\). How many euros do you get for \(\pounds 50\)?

    1. A\(\euro 41.67\)
    2. B\(\euro 60\)
    3. C\(\euro 51.20\)
    4. D\(\euro 600\)
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    B: \(\euro 60\)

    Multiply by the exchange rate: \(50 \times 1.2 = 60\).

  7. 7

    Two taps fill a tank in 30 minutes. How long would 5 identical taps take?

    1. A12 minutes
    2. B75 minutes
    3. C6 minutes
    4. D15 minutes
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    A: 12 minutes

    This is inverse proportion. The total is \(2 \times 30 = 60\) tap-minutes, so 5 taps need \(60 \div 5 = 12\) minutes.

  8. 8

    \(y\) is directly proportional to \(x\). \(y = 12\) when \(x = 3\). Work out \(y\) when \(x = 7\).

    1. A16
    2. B84
    3. C36
    4. D28
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    D: 28

    \(y = kx\) and \(12 = 3k\), so \(k = 4\). Then \(y = 4 \times 7 = 28\).

  9. 9

    \(y\) is inversely proportional to \(x\). \(y = 6\) when \(x = 4\). Work out \(y\) when \(x = 3\).

    1. A4.5
    2. B18
    3. C8
    4. D2
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    C: 8

    \(y = \dfrac{k}{x}\) and \(6 = \dfrac{k}{4}\), so \(k = 24\). Then \(y = \dfrac{24}{3} = 8\).