Mark scheme · Fractions: Four Operations
Total: 20 marks
Award the marks shown for each correct step, then add up the total out of 20. A method mark counts even if the final answer is wrong.
Convert between mixed numbers and improper fractions.
You are given $5\tfrac{1}{2}$ and $\dfrac{17}{3}$.
Write $5\tfrac{1}{2}$ as an improper fraction.[1 mark]
Method$5\tfrac{1}{2} = \dfrac{5 \times 2 + 1}{2} = \dfrac{11}{2}$.
Answer: $\dfrac{11}{2}$
Marks✔1 markImproper fraction $\dfrac{11}{2}$
Work out $\dfrac{1}{2} + \dfrac{5}{12}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodCommon denominator 12: $\dfrac{1}{2} = \dfrac{6}{12}$ and $\dfrac{5}{12} = \dfrac{5}{12}$.
$\dfrac{6}{12} + \dfrac{5}{12} = \dfrac{11}{12}$.
Answer: $\dfrac{11}{12}$
Marks✔1 markCommon denominator 12 ($\dfrac{6}{12}$ and $\dfrac{5}{12}$)
✔1 markAnswer $\dfrac{11}{12}$
Work out $3\tfrac{4}{5} \times \dfrac{1}{6}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodWrite the mixed number as an improper fraction: $3\tfrac{4}{5} = \dfrac{19}{5}$.
Multiply numerators and denominators: $3\tfrac{4}{5} \times \dfrac{1}{6} = \dfrac{19 \times 1}{5 \times 6} = \dfrac{19}{30}$.
Answer: $\dfrac{19}{30}$
Marks✔1 markMultiply tops and bottoms: $\dfrac{19}{30}$
✔1 markAnswer $\dfrac{19}{30}$
A lesson lasts 60 minutes.
Work out $\dfrac{3}{5}$ of the lesson.[2 marks]
Method$\dfrac{1}{5}$ of 60 is 60 ÷ 5 = 12.
$\dfrac{3}{5}$ of 60 is 3 × 12 = 36 minutes.
Answer: $36$
Marks✔1 markDivide by 5: 12
✔1 markMultiply by 3: 36
Convert between mixed numbers and improper fractions.
You are given $3\tfrac{7}{8}$ and $\dfrac{37}{6}$.
Write $3\tfrac{7}{8}$ as an improper fraction.[1 mark]
Method$3\tfrac{7}{8} = \dfrac{3 \times 8 + 7}{8} = \dfrac{31}{8}$.
Answer: $\dfrac{31}{8}$
Marks✔1 markImproper fraction $\dfrac{31}{8}$
Work out $\dfrac{1}{6} + \dfrac{5}{12}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodCommon denominator 12: $\dfrac{1}{6} = \dfrac{2}{12}$ and $\dfrac{5}{12} = \dfrac{5}{12}$.
$\dfrac{2}{12} + \dfrac{5}{12} = \dfrac{7}{12}$.
Answer: $\dfrac{7}{12}$
Marks✔1 markCommon denominator 12 ($\dfrac{2}{12}$ and $\dfrac{5}{12}$)
✔1 markAnswer $\dfrac{7}{12}$
Work out $\dfrac{1}{6} \times \dfrac{3}{5}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodMultiply numerators and denominators: $\dfrac{1}{6} \times \dfrac{3}{5} = \dfrac{1 \times 3}{6 \times 5} = \dfrac{3}{30} = \dfrac{1}{10}$.
Answer: $\dfrac{1}{10}$
Marks✔1 markMultiply tops and bottoms: $\dfrac{3}{30}$
✔1 markAnswer $\dfrac{1}{10}$
A bag holds 15 sweets.
Work out $\dfrac{3}{5}$ of the sweets.[2 marks]
Method$\dfrac{1}{5}$ of 15 is 15 ÷ 5 = 3.
$\dfrac{3}{5}$ of 15 is 3 × 3 = 9 sweets.
Answer: $9$
Marks✔1 markDivide by 5: 3
✔1 markMultiply by 3: 9
Convert between mixed numbers and improper fractions.
You are given $5\tfrac{1}{2}$ and $\dfrac{27}{4}$.
Write $5\tfrac{1}{2}$ as an improper fraction.[1 mark]
Method$5\tfrac{1}{2} = \dfrac{5 \times 2 + 1}{2} = \dfrac{11}{2}$.
Answer: $\dfrac{11}{2}$
Marks✔1 markImproper fraction $\dfrac{11}{2}$
Work out $\dfrac{4}{5} - \dfrac{7}{10}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodCommon denominator 10: $\dfrac{4}{5} = \dfrac{8}{10}$ and $\dfrac{7}{10} = \dfrac{7}{10}$.
$\dfrac{8}{10} - \dfrac{7}{10} = \dfrac{1}{10}$.
Answer: $\dfrac{1}{10}$
Marks✔1 markCommon denominator 10 ($\dfrac{8}{10}$ and $\dfrac{7}{10}$)
✔1 markAnswer $\dfrac{1}{10}$
Work out $2\tfrac{1}{2} \times \dfrac{3}{2}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodWrite the mixed number as an improper fraction: $2\tfrac{1}{2} = \dfrac{5}{2}$.
Multiply numerators and denominators: $2\tfrac{1}{2} \times \dfrac{3}{2} = \dfrac{5 \times 3}{2 \times 2} = \dfrac{15}{4}$.
Answer: $\dfrac{15}{4}$
Marks✔1 markMultiply tops and bottoms: $\dfrac{15}{4}$
✔1 markAnswer $\dfrac{15}{4}$
A lesson lasts 15 minutes.
Work out $\dfrac{3}{5}$ of the lesson.[2 marks]
Method$\dfrac{1}{5}$ of 15 is 15 ÷ 5 = 3.
$\dfrac{3}{5}$ of 15 is 3 × 3 = 9 minutes.
Answer: $9$
Marks✔1 markDivide by 5: 3
✔1 markMultiply by 3: 9
Convert between mixed numbers and improper fractions.
You are given $3\tfrac{1}{2}$ and $\dfrac{14}{5}$.
Write $3\tfrac{1}{2}$ as an improper fraction.[1 mark]
Method$3\tfrac{1}{2} = \dfrac{3 \times 2 + 1}{2} = \dfrac{7}{2}$.
Answer: $\dfrac{7}{2}$
Marks✔1 markImproper fraction $\dfrac{7}{2}$
Work out $\dfrac{1}{6} + \dfrac{11}{12}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodCommon denominator 12: $\dfrac{1}{6} = \dfrac{2}{12}$ and $\dfrac{11}{12} = \dfrac{11}{12}$.
$\dfrac{2}{12} + \dfrac{11}{12} = \dfrac{13}{12}$.
Answer: $\dfrac{13}{12}$
Marks✔1 markCommon denominator 12 ($\dfrac{2}{12}$ and $\dfrac{11}{12}$)
✔1 markAnswer $\dfrac{13}{12}$
Work out $\dfrac{2}{3} \times \dfrac{3}{4}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodMultiply numerators and denominators: $\dfrac{2}{3} \times \dfrac{3}{4} = \dfrac{2 \times 3}{3 \times 4} = \dfrac{6}{12} = \dfrac{1}{2}$.
Answer: $\dfrac{1}{2}$
Marks✔1 markMultiply tops and bottoms: $\dfrac{6}{12}$
✔1 markAnswer $\dfrac{1}{2}$
Work out $1\tfrac{5}{8} + 3\tfrac{2}{3}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodWrite as improper fractions: $1\tfrac{5}{8} = \dfrac{13}{8}$, $3\tfrac{2}{3} = \dfrac{11}{3}$.
Common denominator 24: $\dfrac{39}{24} + \dfrac{88}{24} = \dfrac{127}{24}$.
So the answer is $\dfrac{127}{24} = 5\tfrac{7}{24}$.
Answer: $\dfrac{127}{24}$
Marks✔1 markBoth as improper fractions $\dfrac{13}{8}$ and $\dfrac{11}{3}$
✔1 markCommon denominator and combine: $\dfrac{127}{24}$
✔1 markAnswer $\dfrac{127}{24}$ (= $5\tfrac{7}{24}$)
Work out $\dfrac{5}{6} \div \dfrac{1}{3}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodDividing means multiply by the reciprocal: $\dfrac{5}{6} \div \dfrac{1}{3} = \dfrac{5}{6} \times \dfrac{3}{1}$.
$= \dfrac{15}{6} = \dfrac{5}{2}$.
Answer: $\dfrac{5}{2}$
Marks✔1 markMultiply by the reciprocal: $\dfrac{5}{6}$ x $\dfrac{3}{1}$
✔1 markAnswer $\dfrac{5}{2}$
Work out $1\tfrac{1}{2} + 4\tfrac{1}{4}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodWrite as improper fractions: $1\tfrac{1}{2} = \dfrac{3}{2}$, $4\tfrac{1}{4} = \dfrac{17}{4}$.
Common denominator 4: $\dfrac{6}{4} + \dfrac{17}{4} = \dfrac{23}{4}$.
So the answer is $\dfrac{23}{4} = 5\tfrac{3}{4}$.
Answer: $\dfrac{23}{4}$
Marks✔1 markBoth as improper fractions $\dfrac{3}{2}$ and $\dfrac{17}{4}$
✔1 markCommon denominator and combine: $\dfrac{23}{4}$
✔1 markAnswer $\dfrac{23}{4}$ (= $5\tfrac{3}{4}$)
Work out $\dfrac{3}{4} \div \dfrac{1}{2}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodDividing means multiply by the reciprocal: $\dfrac{3}{4} \div \dfrac{1}{2} = \dfrac{3}{4} \times \dfrac{2}{1}$.
$= \dfrac{6}{4} = \dfrac{3}{2}$.
Answer: $\dfrac{3}{2}$
Marks✔1 markMultiply by the reciprocal: $\dfrac{3}{4}$ x $\dfrac{2}{1}$
✔1 markAnswer $\dfrac{3}{2}$
Work out $3\tfrac{3}{8} + 1\tfrac{4}{5}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodWrite as improper fractions: $3\tfrac{3}{8} = \dfrac{27}{8}$, $1\tfrac{4}{5} = \dfrac{9}{5}$.
Common denominator 40: $\dfrac{135}{40} + \dfrac{72}{40} = \dfrac{207}{40}$.
So the answer is $\dfrac{207}{40} = 5\tfrac{7}{40}$.
Answer: $\dfrac{207}{40}$
Marks✔1 markBoth as improper fractions $\dfrac{27}{8}$ and $\dfrac{9}{5}$
✔1 markCommon denominator and combine: $\dfrac{207}{40}$
✔1 markAnswer $\dfrac{207}{40}$ (= $5\tfrac{7}{40}$)
Work out $\dfrac{2}{3} \div \dfrac{1}{3}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodDividing means multiply by the reciprocal: $\dfrac{2}{3} \div \dfrac{1}{3} = \dfrac{2}{3} \times \dfrac{3}{1}$.
$= \dfrac{6}{3} = 2$.
Answer: $2$
Marks✔1 markMultiply by the reciprocal: $\dfrac{2}{3}$ x $\dfrac{3}{1}$
✔1 markAnswer 2
Work out $1\tfrac{3}{4} + 1\tfrac{2}{5}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodWrite as improper fractions: $1\tfrac{3}{4} = \dfrac{7}{4}$, $1\tfrac{2}{5} = \dfrac{7}{5}$.
Common denominator 20: $\dfrac{35}{20} + \dfrac{28}{20} = \dfrac{63}{20}$.
So the answer is $\dfrac{63}{20} = 3\tfrac{3}{20}$.
Answer: $\dfrac{63}{20}$
Marks✔1 markBoth as improper fractions $\dfrac{7}{4}$ and $\dfrac{7}{5}$
✔1 markCommon denominator and combine: $\dfrac{63}{20}$
✔1 markAnswer $\dfrac{63}{20}$ (= $3\tfrac{3}{20}$)
Work out $\dfrac{2}{3} \div \dfrac{1}{2}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodDividing means multiply by the reciprocal: $\dfrac{2}{3} \div \dfrac{1}{2} = \dfrac{2}{3} \times \dfrac{2}{1}$.
$= \dfrac{4}{3}$.
Answer: $\dfrac{4}{3}$
Marks✔1 markMultiply by the reciprocal: $\dfrac{2}{3}$ x $\dfrac{2}{1}$
✔1 markAnswer $\dfrac{4}{3}$
Work out $2\tfrac{1}{2} + 2\tfrac{2}{3}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodWrite as improper fractions: $2\tfrac{1}{2} = \dfrac{5}{2}$, $2\tfrac{2}{3} = \dfrac{8}{3}$.
Common denominator 6: $\dfrac{15}{6} + \dfrac{16}{6} = \dfrac{31}{6}$.
So the answer is $\dfrac{31}{6} = 5\tfrac{1}{6}$.
Answer: $\dfrac{31}{6}$
Marks✔1 markBoth as improper fractions $\dfrac{5}{2}$ and $\dfrac{8}{3}$
✔1 markCommon denominator and combine: $\dfrac{31}{6}$
✔1 markAnswer $\dfrac{31}{6}$ (= $5\tfrac{1}{6}$)
Work out $3\tfrac{1}{3} \div \dfrac{5}{2}$.
Give your answer as a fraction in its simplest form.[2 marks]
MethodWrite the mixed number as an improper fraction: $3\tfrac{1}{3} = \dfrac{10}{3}$.
Dividing means multiply by the reciprocal: $3\tfrac{1}{3} \div \dfrac{5}{2} = \dfrac{10}{3} \times \dfrac{2}{5}$.
$= \dfrac{20}{15} = \dfrac{4}{3}$.
Answer: $\dfrac{4}{3}$
Marks✔1 markMultiply by the reciprocal: $\dfrac{10}{3}$ x $\dfrac{2}{5}$
✔1 markAnswer $\dfrac{4}{3}$
Work out $3\tfrac{2}{5} + 3\tfrac{5}{6}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodWrite as improper fractions: $3\tfrac{2}{5} = \dfrac{17}{5}$, $3\tfrac{5}{6} = \dfrac{23}{6}$.
Common denominator 30: $\dfrac{102}{30} + \dfrac{115}{30} = \dfrac{217}{30}$.
So the answer is $\dfrac{217}{30} = 7\tfrac{7}{30}$.
Answer: $\dfrac{217}{30}$
Marks✔1 markBoth as improper fractions $\dfrac{17}{5}$ and $\dfrac{23}{6}$
✔1 markCommon denominator and combine: $\dfrac{217}{30}$
✔1 markAnswer $\dfrac{217}{30}$ (= $7\tfrac{7}{30}$)
Work out $\dfrac{1}{2} - \dfrac{1}{2} \times \dfrac{3}{4}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodDo $\times$ / $\div$ first: $\dfrac{1}{2} \times \dfrac{3}{4} = \dfrac{3}{8}$.
Then $\dfrac{1}{2} - \dfrac{3}{8}$: common denominator 8 gives $\dfrac{4}{8} - \dfrac{3}{8} = \dfrac{1}{8}$.
So the answer is $\dfrac{1}{8}$.
Answer: $\dfrac{1}{8}$
Marks✔1 markMultiply/divide first: $\dfrac{3}{8}$
✔1 markCommon denominator and combine
✔1 markAnswer $\dfrac{1}{8}$
Work out $\dfrac{4}{5} + \dfrac{2}{5} \div \dfrac{4}{5}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodDo $\times$ / $\div$ first: $\dfrac{2}{5} \div \dfrac{4}{5} = \dfrac{1}{2}$.
Then $\dfrac{4}{5} + \dfrac{1}{2}$: common denominator 10 gives $\dfrac{8}{10} + \dfrac{5}{10} = \dfrac{13}{10}$.
So the answer is $\dfrac{13}{10}$.
Answer: $\dfrac{13}{10}$
Marks✔1 markMultiply/divide first: $\dfrac{1}{2}$
✔1 markCommon denominator and combine
✔1 markAnswer $\dfrac{13}{10}$
Work out $\dfrac{1}{5} - \dfrac{1}{5} \times \dfrac{3}{5}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodDo $\times$ / $\div$ first: $\dfrac{1}{5} \times \dfrac{3}{5} = \dfrac{3}{25}$.
Then $\dfrac{1}{5} - \dfrac{3}{25}$: common denominator 25 gives $\dfrac{5}{25} - \dfrac{3}{25} = \dfrac{2}{25}$.
So the answer is $\dfrac{2}{25}$.
Answer: $\dfrac{2}{25}$
Marks✔1 markMultiply/divide first: $\dfrac{3}{25}$
✔1 markCommon denominator and combine
✔1 markAnswer $\dfrac{2}{25}$
Work out $\dfrac{1}{6} + \dfrac{1}{5} \times \dfrac{1}{2}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodDo $\times$ / $\div$ first: $\dfrac{1}{5} \times \dfrac{1}{2} = \dfrac{1}{10}$.
Then $\dfrac{1}{6} + \dfrac{1}{10}$: common denominator 30 gives $\dfrac{5}{30} + \dfrac{3}{30} = \dfrac{8}{30}$.
So the answer is $\dfrac{4}{15}$.
Answer: $\dfrac{4}{15}$
Marks✔1 markMultiply/divide first: $\dfrac{1}{10}$
✔1 markCommon denominator and combine
✔1 markAnswer $\dfrac{4}{15}$
Work out $\dfrac{3}{5} - \dfrac{1}{3} \times \dfrac{2}{5}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodDo $\times$ / $\div$ first: $\dfrac{1}{3} \times \dfrac{2}{5} = \dfrac{2}{15}$.
Then $\dfrac{3}{5} - \dfrac{2}{15}$: common denominator 15 gives $\dfrac{9}{15} - \dfrac{2}{15} = \dfrac{7}{15}$.
So the answer is $\dfrac{7}{15}$.
Answer: $\dfrac{7}{15}$
Marks✔1 markMultiply/divide first: $\dfrac{2}{15}$
✔1 markCommon denominator and combine
✔1 markAnswer $\dfrac{7}{15}$
Work out $\dfrac{1}{2} - \dfrac{1}{5} \div \dfrac{1}{2}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodDo $\times$ / $\div$ first: $\dfrac{1}{5} \div \dfrac{1}{2} = \dfrac{2}{5}$.
Then $\dfrac{1}{2} - \dfrac{2}{5}$: common denominator 10 gives $\dfrac{5}{10} - \dfrac{4}{10} = \dfrac{1}{10}$.
So the answer is $\dfrac{1}{10}$.
Answer: $\dfrac{1}{10}$
Marks✔1 markMultiply/divide first: $\dfrac{2}{5}$
✔1 markCommon denominator and combine
✔1 markAnswer $\dfrac{1}{10}$
Work out $\dfrac{1}{2} + \dfrac{3}{4} \div \dfrac{3}{4}$.
Give your answer as a fraction in its simplest form.[3 marks]
MethodDo $\times$ / $\div$ first: $\dfrac{3}{4} \div \dfrac{3}{4} = \dfrac{1}{1}$.
Then $\dfrac{1}{2} + \dfrac{1}{1}$: common denominator 2 gives $\dfrac{1}{2} + \dfrac{2}{2} = \dfrac{3}{2}$.
So the answer is $\dfrac{3}{2}$.
Answer: $\dfrac{3}{2}$
Marks✔1 markMultiply/divide first: $\dfrac{1}{1}$
✔1 markCommon denominator and combine
✔1 markAnswer $\dfrac{3}{2}$
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