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

1Grade 1–3FoundationCalculator2 marks

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

$$0.65 \qquad \frac{5}{8} \qquad \frac{2}{3} \qquad 0.6$$ [2]

for converting at least two of the fractions to decimals (e.g. $\frac58=0.625$, $\frac23=0.66\ldots$) or writing all four with a common denominator, or a correct order with one pair reversedM1
$0.6,\ \frac{5}{8},\ 0.65,\ \frac{2}{3}$ (accept equivalent values in the correct order; accept largest first only if clearly labelled)A1
Answer: $0.6,\ \frac58,\ 0.65,\ \frac23$
Examiner tip: Students who write $\frac23 = 0.6$ cannot separate it from $0.6$ — it is $0.666\ldots$, the largest value.

2Grade 1–3FoundationCalculator3 marks

There are 120 students in Year 9 at a school.

45 of these students walk to school.

(a) Write 45 as a fraction of 120. Give your fraction in its simplest form. [2]

(b) Write your answer to part (a) as a percentage. [1]

(a) for $\frac{45}{120}$ or any correct partial simplification e.g. $\frac{9}{24}$, $\frac{15}{40}$M1
(a) $\frac{3}{8}$A1
(b) $37.5\%$ (ft their fraction from (a) if it is a proper fraction)B1
Answer: (a) $\frac38$ (b) $37.5\%$
Examiner tip: Stopping at $\frac{9}{24}$ or $\frac{15}{40}$ loses the accuracy mark — keep cancelling until no common factor is left.

3Grade 4–5Calculator3 marks

Show that $$2\frac{3}{4} + 1\frac{5}{6} = 4\frac{7}{12}$$ [3]

for converting both to improper fractions $\frac{11}{4}$ and $\frac{11}{6}$, or adding the whole numbers separately ($2+1=3$)M1
for both fractions written with a common denominator, e.g. $\frac{33}{12}+\frac{22}{12}$ or $\frac{9}{12}+\frac{10}{12}$ (any common multiple of 4 and 6)M1
shows working leading to $\frac{55}{12}=4\frac{7}{12}$ or $3+\frac{19}{12}=3+1\frac{7}{12}=4\frac{7}{12}$ (all steps shown; decimals score no marks)A1
Answer: $\frac{33}{12}+\frac{22}{12}=\frac{55}{12}=4\frac{7}{12}$
Examiner tip: In a 'show that' every step must be written: a calculator display or decimals with no fraction working gets no credit.

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4Grade 4–5Calculator3 marks

Show that $$3\frac{1}{3} \div 1\frac{7}{8} = 1\frac{7}{9}$$ [3]

for writing both as improper fractions: $\frac{10}{3}$ and $\frac{15}{8}$M1
for $\frac{10}{3}\times\frac{8}{15}$ (multiplying by the reciprocal of their second fraction) oeM1
shows $\frac{80}{45}=\frac{16}{9}=1\frac{7}{9}$ (or cancels first: $\frac{2}{3}\times\frac{8}{3}=\frac{16}{9}$) with all steps shownA1
Answer: $\frac{10}{3}\times\frac{8}{15}=\frac{16}{9}=1\frac{7}{9}$
Examiner tip: Dividing whole numbers and fractions separately ($3\div1$ and $\frac13\div\frac78$) is wrong and scores nothing — convert to improper fractions first.

5Grade 4–5Calculator4 marks

Amira earns £480 one month.

She spends $\frac{3}{8}$ of the £480 on rent.

She spends $\frac{1}{4}$ of what is left on food.

She saves the rest.

How much money does Amira save? [4]

for $\frac38\times480$ $(=180)$M1
for $480-180$ $(=300)$ or $\frac58\times480$M1
for $300-\frac14\times300$ or $\frac34\times300$ oeM1
£$225$A1
Answer: £225
Examiner tip: The $\frac14$ is of what is LEFT (£300), not of the £480 — finding $\frac14$ of £480 gives the wrong answer of £180.

6Grade 1–3FoundationCalculator2 marks

(a) Write $0.36$ as a fraction in its simplest form. [1]

(b) Write $\frac{7}{40}$ as a decimal. [1]

(a) $\frac{9}{25}$B1
(b) $0.175$B1
Answer: (a) $\frac{9}{25}$ (b) $0.175$
Examiner tip: $\frac{36}{100}$ is correct but not in simplest form, so it does not get the mark.

7Grade 6–7HigherCalculator2 marks

Prove algebraically that the recurring decimal $0.\dot{4}\dot{5}$ can be written as $\frac{5}{11}$ [2]

for $x=0.4545\ldots$ and $100x=45.4545\ldots$ or two other recurring decimals that subtract to give a terminating decimal or integerM1
for $99x=45$ leading to $x=\frac{45}{99}=\frac{5}{11}$ (must show the cancelling)A1
Answer: $100x-x=45 \Rightarrow x=\frac{45}{99}=\frac{5}{11}$
Examiner tip: Dividing 5 by 11 is not a proof — you must set up $x$ and subtract two recurring decimals.

8Grade 8–9HigherCalculator2 marks

Prove algebraically that the recurring decimal $0.1\dot{3}\dot{6}$ can be written as $\frac{3}{22}$ [2]

for $1000x=136.3636\ldots$ and $10x=1.3636\ldots$ (or e.g. $100x=13.6363\ldots$ and $x=0.13636\ldots$) — two recurring decimals that subtract to give a terminating decimalM1
for $990x=135$ leading to $x=\frac{135}{990}=\frac{3}{22}$ (allow $99x=13.5$, $x=\frac{13.5}{99}=\frac{27}{198}=\frac{3}{22}$)A1
Answer: $1000x-10x=135 \Rightarrow x=\frac{135}{990}=\frac{3}{22}$
Examiner tip: Only the 3 and 6 recur; using $100x-x$ leaves $13.5$, so either clear that decimal or use $1000x-10x$.

9Grade 6–7HigherCalculator3 marks

$a = 0.\dot{6}$ and $b = 0.\dot{1}\dot{8}$

Work out $a+b$. Give your answer as a fraction in its simplest form. [3]

for converting one recurring decimal to a fraction: $a=\frac{6}{9}$ or $\frac23$, or $b=\frac{18}{99}$ or $\frac{2}{11}$, or for adding the decimals correctly to get $0.\dot{8}\dot{4}$ $(0.8484\ldots)$M1
for both correct and added using a common denominator, e.g. $\frac{22}{33}+\frac{6}{33}$, or for $0.8484\ldots=\frac{84}{99}$M1
$\frac{28}{33}$A1
Answer: $\frac{28}{33}$
Examiner tip: Adding the decimals as $0.6 + 0.18 = 0.78$ ignores the recurring digits — line them up as $0.6666\ldots + 0.1818\ldots = 0.8484\ldots$

10Grade 6–7HigherCalculator3 marks

The diagram shows a water tank.

The tank is $\frac{3}{5}$ full of water.

84 litres of water are taken out of the tank.

The tank is now $\frac{1}{4}$ full.

Work out the number of litres of water the tank holds when it is full. [3]

for $\frac35-\frac14$ $\left(=\frac{7}{20}\right)$ or $\frac{12}{20}-\frac{5}{20}$M1
for $84\div\frac{7}{20}$ or $84\div7\times20$ oeM1
$240$ (litres)A1
Answer: 240 litres
Examiner tip: 84 litres is the DIFFERENCE between the two levels, so it matches $\frac{7}{20}$ of the tank — not $\frac35$ or $\frac14$.

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