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Exam questions · Maths · Number Without a Calculator

Fractions

  • 7 exam questions
  • 18 marks
  • 10 quick checks
  1. 1 Work out [1 mark]

    Work out \(\dfrac{3}{5}\) of 45.

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    Model answer

    \(45 \div 5 = 9\) and \(9 \times 3 = 27\).

    Mark scheme

    • 27 — B1
  2. 2 Write [2 marks]

    Write \(\dfrac{28}{42}\) in its simplest form.

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    Model answer

    The HCF of 28 and 42 is 14. \(\dfrac{28 \div 14}{42 \div 14} = \dfrac{2}{3}\).

    Mark scheme

    • Divides the top and bottom by a common factor, such as \(\dfrac{14}{21}\) — M1
    • \(\dfrac{2}{3}\) — A1
  3. 3 Work out [3 marks]

    Work out \(\dfrac{2}{3} + \dfrac{4}{5}\). Give your answer as a mixed number.

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    Model answer

    The common denominator is 15: \(\dfrac{10}{15} + \dfrac{12}{15} = \dfrac{22}{15} = 1\dfrac{7}{15}\).

    Mark scheme

    • A common denominator of 15 — M1
    • \(\dfrac{22}{15}\) — A1
    • \(1\dfrac{7}{15}\) — A1
  4. 4 Work out [3 marks]

    Work out \(5\dfrac{1}{2} - 2\dfrac{2}{3}\). Give your answer as a mixed number.

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    Model answer

    \(5\dfrac{1}{2} = \dfrac{11}{2} = \dfrac{33}{6}\) and \(2\dfrac{2}{3} = \dfrac{8}{3} = \dfrac{16}{6}\). Then \(\dfrac{33}{6} - \dfrac{16}{6} = \dfrac{17}{6} = 2\dfrac{5}{6}\).

    Mark scheme

    • Improper fractions or a common denominator of 6 — M1
    • \(\dfrac{33}{6} - \dfrac{16}{6}\) — M1
    • \(2\dfrac{5}{6}\) — A1
  5. 5 Work out [3 marks]

    Work out \(1\dfrac{3}{5} \times 2\dfrac{1}{2}\).

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    Model answer

    \(1\dfrac{3}{5} = \dfrac{8}{5}\) and \(2\dfrac{1}{2} = \dfrac{5}{2}\). Then \(\dfrac{8}{5} \times \dfrac{5}{2} = \dfrac{40}{10} = 4\).

    Mark scheme

    • Both mixed numbers converted to improper fractions — M1
    • \(\dfrac{8}{5} \times \dfrac{5}{2}\) or \(\dfrac{40}{10}\) — M1
    • 4 — A1
  6. 6 Work out [3 marks]

    Work out \(\dfrac{4}{9} \div \dfrac{2}{3}\). Give your answer in its simplest form.

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    Model answer

    \(\dfrac{4}{9} \div \dfrac{2}{3} = \dfrac{4}{9} \times \dfrac{3}{2} = \dfrac{12}{18} = \dfrac{2}{3}\).

    Mark scheme

    • \(\dfrac{4}{9} \times \dfrac{3}{2}\) — M1
    • \(\dfrac{12}{18}\) — A1
    • \(\dfrac{2}{3}\) — A1
  7. 7 Work out [3 marks]

    A tank is \(\dfrac{3}{5}\) full of water. \(\dfrac{1}{3}\) of the water is then used. (a) What fraction of the tank is now full of water? [2 marks] The tank holds 300 litres when it is full. (b) How many litres of water are in the tank now? [1 mark]

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    Model answer

    (a) \(\dfrac{1}{3}\) is used, so \(\dfrac{2}{3}\) of the water is left: \(\dfrac{2}{3} \times \dfrac{3}{5} = \dfrac{6}{15} = \dfrac{2}{5}\). (b) \(\dfrac{2}{5} \times 300 = 120\) litres.

    Mark scheme

    • (a) \(\dfrac{2}{3} \times \dfrac{3}{5}\) or \(\dfrac{3}{5} - \dfrac{1}{3} \times \dfrac{3}{5}\) — M1
    • (a) \(\dfrac{2}{5}\) — A1
    • (b) 120 — B1 (follow through)

Quick check

  1. 1

    \(\dfrac{3}{5}\) of a number is 36. What is the number?

    1. A60
    2. B21.6
    3. C108
    4. D12
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    A: 60

    \(36 \div 3 = 12\) is one fifth, so the whole number is \(12 \times 5 = 60\).

  2. 2

    What fraction of 2 hours is 35 minutes? Give your answer in its simplest form.

    1. A\(\dfrac{35}{2}\)
    2. B\(\dfrac{7}{4}\)
    3. C\(\dfrac{7}{12}\)
    4. D\(\dfrac{7}{24}\)
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    D: \(\dfrac{7}{24}\)

    Convert to minutes: \(\dfrac{35}{120} = \dfrac{7}{24}\).

  3. 3

    Work out \(\dfrac{1}{2} + \dfrac{1}{3}\).

    1. A\(\dfrac{5}{6}\)
    2. B\(\dfrac{2}{6}\)
    3. C\(\dfrac{1}{6}\)
    4. D\(\dfrac{2}{5}\)
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    A: \(\dfrac{5}{6}\)

    Use a common denominator of 6: \(\dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}\).

  4. 4

    What is \(\dfrac{3}{5}\) of 40?

    1. A8
    2. B24
    3. C16
    4. D120
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    B: 24

    \(40 \div 5 = 8\), then \(8 \times 3 = 24\).

  5. 5

    Work out \(\dfrac{2}{3} \div 4\).

    1. A\(\dfrac{8}{3}\)
    2. B6
    3. C\(\dfrac{1}{6}\)
    4. D\(\dfrac{1}{3}\)
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    C: \(\dfrac{1}{6}\)

    Dividing by 4 is multiplying by \(\dfrac{1}{4}\): \(\dfrac{2}{3} \times \dfrac{1}{4} = \dfrac{2}{12} = \dfrac{1}{6}\).

  6. 6

    Write \(3\dfrac{2}{5}\) as an improper fraction.

    1. A\(\dfrac{11}{5}\)
    2. B\(\dfrac{6}{5}\)
    3. C\(\dfrac{17}{5}\)
    4. D\(\dfrac{32}{5}\)
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    C: \(\dfrac{17}{5}\)

    \(3 \times 5 + 2 = 17\), so the fraction is \(\dfrac{17}{5}\).

  7. 7

    Which fraction is equal to \(\dfrac{18}{24}\)?

    1. A\(\dfrac{4}{5}\)
    2. B\(\dfrac{2}{3}\)
    3. C\(\dfrac{9}{10}\)
    4. D\(\dfrac{3}{4}\)
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    D: \(\dfrac{3}{4}\)

    The HCF of 18 and 24 is 6, and dividing both by 6 gives \(\dfrac{3}{4}\).

  8. 8

    Work out \(\dfrac{3}{4} \times \dfrac{2}{9}\).

    1. A\(\dfrac{27}{8}\)
    2. B\(\dfrac{5}{13}\)
    3. C\(\dfrac{2}{3}\)
    4. D\(\dfrac{1}{6}\)
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    D: \(\dfrac{1}{6}\)

    Multiply across and simplify: \(\dfrac{6}{36} = \dfrac{1}{6}\).

  9. 9

    What is the reciprocal of \(\dfrac{5}{7}\)?

    1. A\(\dfrac{5}{7}\)
    2. B\(\dfrac{7}{5}\)
    3. C\(-\dfrac{5}{7}\)
    4. D35
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    B: \(\dfrac{7}{5}\)

    The reciprocal is the fraction turned upside down, which is \(\dfrac{7}{5}\).

  10. 10

    Which of these fractions is the largest?

    1. A\(\dfrac{2}{3}\)
    2. B\(\dfrac{5}{8}\)
    3. C\(\dfrac{3}{5}\)
    4. D\(\dfrac{7}{12}\)
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    A: \(\dfrac{2}{3}\)

    As decimals they are 0.625, 0.667, 0.583 and 0.6, so \(\dfrac{2}{3}\) is the largest.