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Exam questions · Maths · Ratio and Proportion

Direct and Inverse Proportion

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

    A recipe for 12 biscuits uses 180 g of flour, 90 g of sugar and 60 g of butter. Calculate the amount of each ingredient needed for 30 biscuits.

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    Model answer

    The scale factor is \(30 \div 12 = 2.5\). Flour: \(180 \times 2.5 = 450\) g. Sugar: \(90 \times 2.5 = 225\) g. Butter: \(60 \times 2.5 = 150\) g.

    Mark scheme

    • Scale factor 2.5, or the amounts for 6 biscuits found — M1
    • Two of the three amounts correct — A1
    • 450 g, 225 g and 150 g — A1
  2. 2 Calculate [2 marks]

    8 pencils cost \(\pounds 5.20\). Calculate the cost of 5 pencils.

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    Model answer

    One pencil costs \(5.20 \div 8 = \pounds 0.65\), so 5 pencils cost \(5 \times 0.65 = \pounds 3.25\).

    Mark scheme

    • \(5.20 \div 8 = 0.65\) — M1
    • \(\pounds 3.25\) — A1
  3. 3 Calculate [3 marks]

    A shop sells potatoes in two bags. 3 kg bag: \(\pounds 2.40\) 5 kg bag: \(\pounds 3.75\) Which bag is the better value for money? You must show your working.

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    Model answer

    3 kg bag: \(2.40 \div 3 = \pounds 0.80\) per kg. 5 kg bag: \(3.75 \div 5 = \pounds 0.75\) per kg. The 5 kg bag is cheaper per kilogram, so it is better value.

    Mark scheme

    • A method to compare, such as the cost per kg — M1
    • \(\pounds 0.80\) and \(\pounds 0.75\) per kg, or equivalent — A1
    • 5 kg bag, with a correct comparison — A1
  4. 4 Calculate [2 marks]

    The exchange rate is \(\pounds 1 = \$1.30\). Calculate how many pounds you get for \(\$390\).

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    Model answer

    \(390 \div 1.3 = 300\), so you get \(\pounds 300\).

    Mark scheme

    • \(390 \div 1.3\) — M1
    • \(\pounds 300\) — A1
  5. 5 Calculate [2 marks]

    6 people take 8 hours to build a shed. All the people work at the same rate. Calculate how long it takes 4 people to build the shed.

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    Model answer

    The job needs \(6 \times 8 = 48\) person-hours, and \(48 \div 4 = 12\) hours.

    Mark scheme

    • \(6 \times 8 = 48\) — M1
    • 12 hours — A1
  6. 6 Find [4 marks]

    \(y\) is directly proportional to \(x\). When \(x = 4\), \(y = 10\). (a) Find a formula for \(y\) in terms of \(x\). [2 marks] (b) Find the value of \(x\) when \(y = 25\). [2 marks]

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    Model answer

    (a) \(y = kx\), so \(10 = 4k\) and \(k = 2.5\). The formula is \(y = 2.5x\). (b) \(25 = 2.5x\), so \(x = 25 \div 2.5 = 10\).

    Mark scheme

    • (a) \(y = kx\) with \(k = 2.5\) found — M1
    • (a) \(y = 2.5x\) — A1
    • (b) \(25 \div 2.5\) — M1
    • (b) 10 — 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 \(y = \dfrac{24}{3} = 8\).