Fractions, decimals & recurring decimals – GCSE Maths exam questions with mark scheme

1Grade 1–3FoundationNon-calculator2 marks

Work out $\dfrac{2}{3} + \dfrac{1}{5}$

Give your answer as a fraction in its simplest form. [2]

for both fractions written with a common denominator, e.g. $\frac{10}{15} + \frac{3}{15}$M1
$\frac{13}{15}$A1
Answer: $\frac{13}{15}$
Examiner tip: Never add the denominators: 2/3 + 1/5 is not 3/8.

2Grade 1–3FoundationNon-calculator2 marks

(a) Write $0.45$ as a fraction in its simplest form. [1]

(b) Write $\dfrac{7}{8}$ as a decimal. [1]

(a) $\frac{9}{20}$B1
(b) $0.875$B1
Answer: (a) $\frac{9}{20}$ (b) 0.875
Examiner tip: 'Simplest form' is required: 45/100 alone does not score in (a).

3Grade 4–5Non-calculator4 marks

Work out

(a) $2\dfrac{1}{4} \times 1\dfrac{3}{5}$ [2]

(b) $2\dfrac{2}{5} \div \dfrac{3}{4}$ [2]

Give each answer as a mixed number in its simplest form.

(a) for $\frac{9}{4} \times \frac{8}{5}$ (both improper fractions correct)M1
(a) $3\frac{3}{5}$A1
(b) for $\frac{12}{5} \times \frac{4}{3}$ (improper fraction and reciprocal)M1
(b) $3\frac{1}{5}$A1
Answer: (a) $3\frac{3}{5}$ (b) $3\frac{1}{5}$
Examiner tip: Convert to improper fractions first; multiplying whole parts and fraction parts separately is wrong.

4Grade 4–5Non-calculator3 marks

There are 840 students in a school.

$\dfrac{3}{8}$ of the students walk to school.

$\dfrac{3}{7}$ of the remaining students travel by bus.

The rest of the students travel by car.

How many students travel by car? [3]

for $840 \div 8 \times 3 = 315$ (walk) or $840 \times \frac{5}{8} = 525$ (remaining)P1
for $\frac{3}{7} \times 525 = 225$ (bus) or $\frac{4}{7} \times 525$P1
$300$A1
Answer: 300
Examiner tip: 'Of the remaining' means take 3/7 of 525, not of 840 – using 840 gives 165, which scores only the first mark.

5Grade 6–7HigherNon-calculator2 marks

Here are four fractions.

$\dfrac{3}{20}\qquad \dfrac{5}{12}\qquad \dfrac{7}{35}\qquad \dfrac{9}{24}$

Without working out the decimals, decide which of these fractions are equivalent to recurring decimals. Give a reason for your answer. [2]

$\frac{5}{12}$ only identified as recurringB1
for a reason referring to prime factors of the denominator in simplest form, e.g. $12 = 2^2 \times 3$ has a prime factor 3 (other than 2 or 5), whereas $20 = 2^2 \times 5$, $\frac{7}{35} = \frac{1}{5}$ and $\frac{9}{24} = \frac{3}{8}$ have denominators with only 2s and 5sC1
Answer: $\frac{5}{12}$ only
Examiner tip: Simplify first: 9/24 = 3/8 terminates even though 24 has a factor of 3.

6Grade 6–7HigherNon-calculator2 marks

Prove algebraically that the recurring decimal $0.\dot{2}\dot{7}$ can be written as $\dfrac{3}{11}$ [2]

for $x = 0.2727…$ and $100x = 27.2727…$ and subtracting (or equivalent pair of recurring decimals that differ by an integer)M1
for $99x = 27$ leading to $x = \frac{27}{99} = \frac{3}{11}$ with complete workingA1
Answer: $99x = 27 \Rightarrow x = \frac{27}{99} = \frac{3}{11}$
Examiner tip: Show the recurring dots or '…' on both lines and show 27/99 cancelling to 3/11 – a 'prove' question needs every step, ending with the given fraction.

7Grade 8–9HigherNon-calculator3 marks

Express the recurring decimal $0.3\dot{1}\dot{8}$ as a fraction in its simplest form. [3]

for two correct recurring decimals that differ by an integer, e.g. $x = 0.31818…$, $10x = 3.1818…$ and $1000x = 318.1818…$M1
for subtracting to give $990x = 315$ or $x = \frac{315}{990}$ oe (e.g. $\frac{31.5}{99}$)M1
$\frac{7}{22}$A1
Answer: $\frac{7}{22}$
Examiner tip: Only '18' repeats, so subtract 10x from 1000x; using 1000x − x leaves a non-recurring tail.

8Grade 1–3FoundationNon-calculator3 marks

(a) The diagram shows a rectangle divided into equal squares.

What fraction of the rectangle is shaded? Give your answer in its simplest form. [1]

(b) Write 40 cm as a fraction of 3 m.

Give your answer in its simplest form. [2]

(a) $\dfrac{2}{5}$ cao (accept $\frac{8}{20}$ or $\frac{4}{10}$ seen with final answer $\frac{2}{5}$; $\frac{8}{20}$ alone scores B0)B1
(b) for converting to the same units, e.g. 3 m = 300 cm, or $\dfrac{40}{300}$ oe (e.g. $\frac{0.4}{3}$)M1
(b) $\dfrac{2}{15}$ caoA1
Answer: (a) $\frac{2}{5}$ (b) $\frac{2}{15}$
Examiner tip: Change both quantities to the same unit first – writing $\frac{40}{3}$ mixes centimetres and metres and scores nothing.

9Grade 1–3FoundationNon-calculator2 marks

Write these numbers in order of size. Start with the smallest number. [2]

$0.62\qquad \dfrac{5}{8}\qquad \dfrac{13}{20}\qquad 0.605\qquad \dfrac{3}{5}$

for converting at least two of the fractions to decimals (0.625, 0.65, 0.6) or all five to a common form (e.g. thousandths 620, 625, 650, 605, 600)M1
$\dfrac{3}{5},\ 0.605,\ 0.62,\ \dfrac{5}{8},\ \dfrac{13}{20}$ (accept equivalent values in place of the originals; correct order but largest first scores M1 A0)A1
Answer: $\frac{3}{5},\ 0.605,\ 0.62,\ \frac{5}{8},\ \frac{13}{20}$
Examiner tip: Write every number with the same number of decimal places (0.600, 0.605, 0.620, 0.625, 0.650) – students often think 0.605 is bigger than 0.62 because 605 > 62.

10Grade 4–5Non-calculator4 marks

Work out

(a) $2.8 \times 0.35$ [2]

(b) $15.12 \div 0.4$ [2]

(a) for a complete method, e.g. $28 \times 35 = 980$ by any correct written method (grid, column, lattice) with no more than one arithmetic errorM1
(a) $0.98$ cao (accept 0.980)A1
(b) for rewriting as an equivalent division with an integer divisor, e.g. $151.2 \div 4$ or $1512 \div 40$, or for digits 378 seenM1
(b) $37.8$ caoA1
Answer: (a) 0.98 (b) 37.8
Examiner tip: In (b) multiply BOTH numbers by 10 to get $151.2 \div 4$; dividing by 0.4 makes the answer bigger, so 3.78 cannot be right.

11Grade 4–5Non-calculator3 marks

Work out $3\dfrac{1}{6} - 4\dfrac{3}{4}$

Give your answer as a mixed number in its simplest form. [3]

for writing both as improper fractions, $\dfrac{19}{6}$ and $\dfrac{19}{4}$, or for dealing with whole numbers and fractions separately, e.g. $-1$ and $\dfrac{1}{6} - \dfrac{3}{4}$M1
for both fractions written over a common denominator, e.g. $\dfrac{38}{12} - \dfrac{57}{12}$ or $\dfrac{2}{12} - \dfrac{9}{12}$M1
$-1\dfrac{7}{12}$ cao (do not accept $-\frac{19}{12}$ as the final answer; $-\frac{5}{12}$ from $-1+\frac{7}{12}$ scores M1 M1 A0)A1
Answer: $-1\frac{7}{12}$
Examiner tip: The answer is negative: $-1$ and $-\frac{7}{12}$ combine to make $-1\frac{7}{12}$, not $-\frac{5}{12}$ – converting to improper fractions avoids this trap.

12Grade 4–5Non-calculator4 marks

A farm has an area of 160 hectares.

The table shows the fraction of the farm used for each purpose.

Use	Fraction of farm
Wheat	$\dfrac{3}{8}$
Barley	$\dfrac{1}{5}$
Woodland	$0.15$
Grass

(a) What fraction of the farm is used for grass? [2]

(b) Work out how many more hectares are used for barley than for woodland. [1]

Is Ravi correct? You must show how you decide. [1]

(a) for writing the fractions with a common denominator or as decimals and subtracting from 1, e.g. $1 - \dfrac{15}{40} - \dfrac{8}{40} - \dfrac{6}{40}$ or $1 - 0.375 - 0.2 - 0.15$; or for finding the hectares 60, 32 and 24 and $160 - 116 = 44$M1
(a) $\dfrac{11}{40}$ oe (accept 0.275 or $\frac{44}{160}$)A1
(b) 8 (hectares) from $32 - 24$B1
(c) Yes, with a correct comparison, e.g. $\frac{11}{40} > \frac{10}{40}$, or $0.275 > 0.25$, or 44 ha > 40 ha (ft their answer to (a) if it is a fraction less than 1)C1
Answer: (a) $\frac{11}{40}$ (b) 8 hectares (c) Yes – $\frac{11}{40} > \frac{10}{40}$
Examiner tip: Change 0.15 into a fraction ($\frac{3}{20} = \frac{6}{40}$) before subtracting – mixing decimals and fractions in one line is where most errors happen.

13Grade 6–7HigherNon-calculator4 marks

(a) Write $\dfrac{4}{7}$ as a recurring decimal. Use dot notation in your answer. [2]

(b) Work out the 50th digit after the decimal point when $\dfrac{4}{7}$ is written as a decimal. [2]

(a) for a correct short division of 4 by 7 giving at least 0.5714 (with remainders that repeat)M1
(a) $0.\dot{5}7142\dot{8}$ (dots on the 5 and the 8 only; accept a bar over 571428; 0.571428... without correct notation scores M1 A0)A1
(b) for recognising the repeating block has 6 digits and using $50 = 6 \times 8 + 2$ oe (e.g. 48th digit is 8)M1
(b) 7 caoA1
Answer: (a) $0.\dot{5}7142\dot{8}$ (b) 7
Examiner tip: For a block of repeating digits, dots go on the FIRST and LAST digits of the block only – putting a dot on every digit, or only on the 8, loses the accuracy mark.

14Grade 6–7HigherNon-calculator3 marks

Work out $0.\dot{4} + 0.1\dot{6}$

Give your answer as a fraction in its simplest form. [3]

for converting one recurring decimal correctly, e.g. $0.\dot{4} = \dfrac{4}{9}$, or $0.1\dot{6} = \dfrac{15}{90}$ oe (e.g. $\frac{1.5}{9}$, $\frac{1}{6}$) using a valid method such as $100x - 10x = 15$M1
for both converted correctly and written over a common denominator, e.g. $\dfrac{8}{18} + \dfrac{3}{18}$ or $\dfrac{40}{90} + \dfrac{15}{90}$M1
$\dfrac{11}{18}$ cao (an unsimplified correct answer such as $\frac{55}{90}$ scores M1 M1 A0)A1
Answer: $\frac{11}{18}$
Examiner tip: In $0.1\dot{6}$ only the 6 recurs, so it is $\frac{1}{6}$, not $\frac{16}{99}$ – check by writing out 0.1666… before choosing which multiples of $x$ to subtract.

15Grade 8–9HigherNon-calculator4 marks

Work out $0.\dot{6} \times 0.\dot{4}\dot{5}$

Give your answer as a recurring decimal. [4]

for $0.\dot{6} = \dfrac{6}{9}$ oe (e.g. $\frac{2}{3}$)M1
for $0.\dot{4}\dot{5} = \dfrac{45}{99}$ oe (e.g. $\frac{5}{11}$), e.g. from $100x - x = 45$M1
for multiplying their fractions correctly to get $\dfrac{10}{33}$ oe (e.g. $\frac{270}{891}$, $\frac{30}{99}$)M1
$0.\dot{3}\dot{0}$ cao (accept 0.303030… with a bar over 30; 0.30 with no recurring notation scores A0)A1
Answer: $0.\dot{3}\dot{0}$
Examiner tip: Do not multiply the decimals as if they ended (0.6 × 0.45 = 0.27 scores 0) – convert both to fractions first, then convert $\frac{10}{33} = \frac{30}{99}$ back to a recurring decimal.

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