Fractions

1. 1 Put [2 marks]

   Put these fractions in order, smallest first. \(\dfrac{5}{8}\) \(\dfrac{3}{5}\) \(\dfrac{7}{10}\)

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   Model answer

   Use a common denominator of 40: \(\dfrac{25}{40}\), \(\dfrac{24}{40}\) and \(\dfrac{28}{40}\). The order is \(\dfrac{3}{5},\ \dfrac{5}{8},\ \dfrac{7}{10}\).

   Mark scheme

   • Converts to a common denominator, or to decimals (0.625, 0.6, 0.7) — M1
   • \(\dfrac{3}{5}, \dfrac{5}{8}, \dfrac{7}{10}\) — A1
2. 2 Work out [2 marks]

   Work out \(\dfrac{5}{6} - \dfrac{3}{8}\).

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   Model answer

   The LCM of 6 and 8 is 24. \(\dfrac{20}{24} - \dfrac{9}{24} = \dfrac{11}{24}\).

   Mark scheme

   • \(\dfrac{20}{24}\) and \(\dfrac{9}{24}\) — M1
   • \(\dfrac{11}{24}\) — A1
3. 3 Work out [3 marks]

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

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   Model answer

   \(1\dfrac{2}{3} = \dfrac{5}{3} = \dfrac{25}{15}\) and \(2\dfrac{3}{5} = \dfrac{13}{5} = \dfrac{39}{15}\). Adding gives \(\dfrac{64}{15} = 4\dfrac{4}{15}\).

   Mark scheme

   • A common denominator of 15, or whole numbers and fractions dealt with separately using a common denominator — M1
   • \(\dfrac{25}{15} + \dfrac{39}{15}\) or \(3 + \dfrac{10}{15} + \dfrac{9}{15}\) — M1
   • \(4\dfrac{4}{15}\) — A1
4. 4 Work out [3 marks]

   Work out \(3\dfrac{3}{4} \times 2\dfrac{2}{5}\).

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   Model answer

   \(3\dfrac{3}{4} = \dfrac{15}{4}\) and \(2\dfrac{2}{5} = \dfrac{12}{5}\). Cancel: \(\dfrac{15}{4} \times \dfrac{12}{5} = \dfrac{3}{1} \times \dfrac{3}{1} = 9\).

   Mark scheme

   • Both mixed numbers converted to improper fractions — M1
   • \(\dfrac{15}{4} \times \dfrac{12}{5}\) or \(\dfrac{180}{20}\) — M1
   • 9 — A1
5. 5 Work out [3 marks]

   Work out \(2\dfrac{1}{4} \div 1\dfrac{1}{8}\).

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   Model answer

   \(2\dfrac{1}{4} = \dfrac{9}{4}\) and \(1\dfrac{1}{8} = \dfrac{9}{8}\). Then \(\dfrac{9}{4} \div \dfrac{9}{8} = \dfrac{9}{4} \times \dfrac{8}{9} = 2\).

   Mark scheme

   • Both mixed numbers converted to improper fractions — M1
   • \(\dfrac{9}{4} \times \dfrac{8}{9}\) — M1
   • 2 — A1
6. 6 Work out [3 marks]

   In a class of 30 students, \(\dfrac{2}{5}\) walk to school and \(\dfrac{1}{3}\) come by bus. The rest are driven. How many students are driven to school? You must show your working.

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   Model answer

   Walk: \(\dfrac{2}{5} \times 30 = 12\). Bus: \(\dfrac{1}{3} \times 30 = 10\). Driven: \(30 - 12 - 10 = 8\) students.

   Mark scheme

   • \(\dfrac{2}{5} \times 30 = 12\) or \(\dfrac{1}{3} \times 30 = 10\) — M1
   • \(30 - 12 - 10\) — M1
   • 8 — A1
7. 7 Explain [3 marks]

   Lena says that \(\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{2}{5}\). (a) Explain what Lena has done wrong. [1 mark] (b) Work out the correct answer. [2 marks]

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   Model answer

   (a) Lena has added the numerators and added the denominators, instead of finding a common denominator. (b) \(\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{3}{6} + \dfrac{2}{6} = \dfrac{5}{6}\).

   Mark scheme

   • (a) States that she added the numerators and the denominators — B1
   • (b) \(\dfrac{3}{6}\) and \(\dfrac{2}{6}\) — M1
   • (b) \(\dfrac{5}{6}\) — A1

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.