Fractions

1. 1 Work out [2 marks]

   What fraction of the shape is shaded? Give your answer in its simplest form.

   

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   Model answer

   There are 20 equal squares and 12 are shaded, so the fraction is \(\dfrac{12}{20}\). Dividing the top and bottom by 4 gives \(\dfrac{3}{5}\).

   Mark scheme

   • \(\dfrac{12}{20}\) or 12 shaded out of 20 — M1
   • \(\dfrac{3}{5}\) — A1
2. 2 Work out [3 marks]

   Work out \(\dfrac{3}{4} + \dfrac{5}{6}\). Give your answer as a mixed number.

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   Model answer

   A common denominator is 12: \(\dfrac{9}{12} + \dfrac{10}{12} = \dfrac{19}{12} = 1\dfrac{7}{12}\).

   Mark scheme

   • A common denominator of 12 (or a multiple of 12) — M1
   • \(\dfrac{9}{12} + \dfrac{10}{12}\) or \(\dfrac{19}{12}\) — M1
   • \(1\dfrac{7}{12}\) — A1
3. 3 Work out [3 marks]

   Work out \(4\dfrac{1}{5} - 2\dfrac{3}{4}\). Give your answer as a mixed number.

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   Model answer

   \(4\dfrac{1}{5} = \dfrac{21}{5} = \dfrac{84}{20}\) and \(2\dfrac{3}{4} = \dfrac{11}{4} = \dfrac{55}{20}\). So \(\dfrac{84}{20} - \dfrac{55}{20} = \dfrac{29}{20} = 1\dfrac{9}{20}\).

   Mark scheme

   • Converts to improper fractions or to a common denominator of 20 — M1
   • \(\dfrac{84}{20} - \dfrac{55}{20}\) or equivalent — M1
   • \(1\dfrac{9}{20}\) — A1
4. 4 Work out [3 marks]

   Work out \(2\dfrac{1}{4} \times 1\dfrac{1}{3}\). Give your answer as a mixed number or integer.

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   Model answer

   \(2\dfrac{1}{4} = \dfrac{9}{4}\) and \(1\dfrac{1}{3} = \dfrac{4}{3}\). Then \(\dfrac{9}{4} \times \dfrac{4}{3} = \dfrac{36}{12} = 3\).

   Mark scheme

   • Converts both mixed numbers to improper fractions — M1
   • \(\dfrac{9}{4} \times \dfrac{4}{3}\) or \(\dfrac{36}{12}\) — M1
   • 3 — A1
5. 5 Work out [3 marks]

   Work out \(\dfrac{5}{8} \div \dfrac{3}{4}\). Give your answer in its simplest form.

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   Model answer

   \(\dfrac{5}{8} \div \dfrac{3}{4} = \dfrac{5}{8} \times \dfrac{4}{3} = \dfrac{20}{24} = \dfrac{5}{6}\).

   Mark scheme

   • Inverts the second fraction: \(\dfrac{5}{8} \times \dfrac{4}{3}\) — M1
   • \(\dfrac{20}{24}\) — M1
   • \(\dfrac{5}{6}\) — A1
6. 6 Work out [4 marks]

   Priya earns \(\pounds 480\) one week. She spends \(\dfrac{3}{8}\) of it on rent. She spends \(\dfrac{1}{4}\) of the money she has left on food. How much does she have left after paying for rent and food?

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   Model answer

   Rent: \(\dfrac{3}{8} \times 480 = 60 \times 3 = \pounds 180\). Left after rent: \(480 - 180 = \pounds 300\). Food: \(\dfrac{1}{4} \times 300 = \pounds 75\). Left: \(300 - 75 = \pounds 225\).

   Mark scheme

   • \(\dfrac{3}{8} \times 480 = 180\) — M1
   • \(480 - 180 = 300\) — M1
   • \(\dfrac{1}{4} \times 300 = 75\) — M1
   • \(\pounds 225\) — A1
7. 7 Show that [3 marks]

   Show that \(1\dfrac{2}{5} \div \dfrac{7}{10} = 2\)

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   Model answer

   \(1\dfrac{2}{5} = \dfrac{7}{5}\), so \(\dfrac{7}{5} \div \dfrac{7}{10} = \dfrac{7}{5} \times \dfrac{10}{7} = \dfrac{70}{35} = 2\), as required.

   Mark scheme

   • Converts \(1\dfrac{2}{5}\) to \(\dfrac{7}{5}\) — M1
   • Inverts to give \(\dfrac{7}{5} \times \dfrac{10}{7}\) — M1
   • Cancels or evaluates \(\dfrac{70}{35}\) and states the result is 2 — C1

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.