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Exam questions · Maths · Ratio and Proportion

Direct and Inverse Proportion

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

    Here are the ingredients for 4 people: 300 g of rice, 120 g of peas and 2 onions. Work out the amounts needed for 10 people.

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    Model answer

    The scale factor is \(10 \div 4 = 2.5\). Rice: \(300 \times 2.5 = 750\) g. Peas: \(120 \times 2.5 = 300\) g. Onions: \(2 \times 2.5 = 5\).

    Mark scheme

    • Scale factor 2.5, or the amounts for 2 people found — M1
    • Two of the three amounts correct — A1
    • 750 g, 300 g and 5 onions — A1
  2. 2 Work out [2 marks]

    7 notebooks cost \(\pounds 10.50\). Work out the cost of 12 of these notebooks.

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    Model answer

    One notebook costs \(10.50 \div 7 = \pounds 1.50\), so 12 cost \(12 \times 1.50 = \pounds 18\).

    Mark scheme

    • \(10.50 \div 7 = 1.50\) — M1
    • \(\pounds 18\) — A1
  3. 3 Work out [3 marks]

    Here are two offers for tea bags. Offer A (80 tea bags) costs \(\pounds 2.40\) Offer B (240 tea bags) costs \(\pounds 6.60\) Which offer is better value for money? You must show your working.

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    Model answer

    Offer A: \(240 \div 80 = 3\) pence per tea bag. Offer B: \(660 \div 240 = 2.75\) pence per tea bag. Offer B is cheaper per tea bag, so it is better value.

    Mark scheme

    • A method to compare, such as the cost of one tea bag or the cost of 240 tea bags for offer A — M1
    • 3p and 2.75p, or \(\pounds 7.20\) and \(\pounds 6.60\) — A1
    • Offer B, with a correct comparison — Q1
  4. 4 Work out [2 marks]

    \(\pounds 1 = \euro 1.25\). Work out how many pounds you get for \(\euro 200\).

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    Model answer

    \(200 \div 1.25 = 160\), so you get \(\pounds 160\).

    Mark scheme

    • \(200 \div 1.25\) — M1
    • \(\pounds 160\) — A1
  5. 5 Work out [2 marks]

    5 machines take 12 hours to make a batch of parts. All the machines work at the same rate. How long would 4 machines take to make the same batch?

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    Model answer

    The batch needs \(5 \times 12 = 60\) machine-hours, and \(60 \div 4 = 15\) hours.

    Mark scheme

    • \(5 \times 12 = 60\) — M1
    • 15 hours — A1
  6. 6 Work out [4 marks]

    \(y\) is directly proportional to the square of \(x\). When \(x = 3\), \(y = 18\). (a) Work out the value of \(y\) when \(x = 5\). (3 marks) (b) Work out the positive value of \(x\) when \(y = 32\). (1 mark)

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    Model answer

    (a) \(y = kx^2\), so \(18 = k \times 9\) and \(k = 2\). When \(x = 5\), \(y = 2 \times 25 = 50\). (b) \(32 = 2x^2\), so \(x^2 = 16\) and \(x = 4\).

    Mark scheme

    • (a) \(y = kx^2\) with \(k = 2\) found — M1
    • (a) \(2 \times 5^2\) — M1
    • (a) 50 — A1
    • (b) 4 — B1

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 \(y = \dfrac{24}{3} = 8\).