Fractions and recurring decimals
Fractions and recurring decimals
- Unit 1: Integers
- Unit 2: Place value & Rounding
- Unit 3: Decimals
- Unit 4: Fractions
- Unit 5: Percentages
- Unit 6: Ratio & Proportion
🎯 In this topic you will
- Recognise fractions that are equivalent to recurring decimals
- Deduce whether fractions have recurring or terminating decimal equivalents
🧠 Key Words
- equivalent decimal
- improper fraction
- mixed number
- recurring decimal
- terminating decimal
- unit fraction
Show Definitions
- equivalent decimal: A decimal number that represents the same value as another fraction or decimal.
- improper fraction: A fraction where the numerator is equal to or larger than the denominator.
- mixed number: A number consisting of a whole number and a fraction together.
- recurring decimal: A decimal in which one or more digits repeat infinitely (e.g., 0.333...).
- terminating decimal: A decimal that ends after a finite number of digits (e.g., 0.25).
- unit fraction: A fraction with numerator 1 and any positive integer as the denominator.
You already know how to use equivalent fractions to convert a fraction with a denominator that is a factor of 10 or 100 to a decimal.
For example: $\tfrac{3}{5}=\tfrac{6}{10}=0.6$ and $\tfrac{3}{20}=\tfrac{15}{100}=0.15$
You can also use division to convert a fraction to an equivalent decimal.
The fraction $\tfrac{5}{8}$ is ‘five eighths’, ‘five out of eight’ or ‘five divided by eight’.
To work out the fraction as a decimal, divide 5 by 8: $5 \div 8 = 0.625$
The decimal $0.625$ is a terminating decimal because it comes to an end.
When you convert the fraction $\tfrac{71}{99}$ to a decimal you get: $71 \div 99 = 0.71717171\ldots$
The number $0.71717171\ldots$ is a recurring decimal as the digits 7 and 1 carry on repeating forever.
🔎 Reasoning Tip
Methods: You can use a written method or a calculator to do this.
You can write $0.71717171\ldots$ with the three dots at the end to show that the number goes on forever.
You can also write the number as $0.\overline{71}$, or using dotted notation as $0.\dot{7}\dot{1}$, to show that the 7 and 1 carry on repeating forever.
When you convert the fraction $\tfrac{1}{14}$ to a decimal, you get $1 \div 14 = 0.0714285714285714285\ldots$
🔎 Reasoning Tip
Recurring decimals: A recurring decimal can always be written as a fraction.
You can see that $714285$ in the decimal is repeating, so you write this as $0.\overline{714285}$.
You can also show the repeating block with dots above the first and last digits of the block: $0.\dot{7}14285\dot{5}$. You put a dot above the 7 and the 5 to show that all the digits from 7 to 5 are repeated.
❓ EXERCISES
1. Use a written method to convert these unit fractions into decimals.
Write if the fraction is a terminating or recurring decimal.
The first two have been done for you.
🔎 Reasoning Tip
Unit fractions: A unit fraction has a numerator of 1, e.g. \( \tfrac{1}{2}, \tfrac{1}{3}, \tfrac{1}{4}, \dots \).
a. $\dfrac{1}{2}$ b. $\dfrac{1}{3}$ c. $\dfrac{1}{4}$ d. $\dfrac{1}{5}$
e. $\dfrac{1}{6}$ f. $\dfrac{1}{7}$ g. $\dfrac{1}{8}$
h. $\dfrac{1}{9}$ i. $\dfrac{1}{10}$ j. $\dfrac{1}{11}$ k. $\dfrac{1}{12}$
🔎 Reasoning Tip
In part f, you will need to keep going with the division for quite a long time!
👀 Show answer
a. $\dfrac{1}{2}=0.5$ — terminating
b. $\dfrac{1}{3}=0.\overline{3}$ — recurring
c. $\dfrac{1}{4}=0.25$ — terminating
d. $\dfrac{1}{5}=0.2$ — terminating
e. $\dfrac{1}{6}=0.1\overline{6}$ — recurring
f. $\dfrac{1}{7}=0.\overline{142857}$ — recurring
g. $\dfrac{1}{8}=0.125$ — terminating
h. $\dfrac{1}{9}=0.\overline{1}$ — recurring
i. $\dfrac{1}{10}=0.1$ — terminating
j. $\dfrac{1}{11}=0.\overline{09}$ — recurring
k. $\dfrac{1}{12}=0.08\overline{3}$ — recurring
🧠 Think like a Mathematician
2. Work with a partner or in a small group to answer these questions.
a Copy and complete this table. Use your answers to Question 1.
| Unit fraction | $\frac{1}{2}$ | $\frac{1}{3}$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{1}{6}$ | $\frac{1}{7}$ | $\frac{1}{8}$ | $\frac{1}{9}$ | $\frac{1}{10}$ | $\frac{1}{11}$ | $\frac{1}{12}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Decimal | $0.5$ | $0.\overline{3}$ | |||||||||
| Terminating (T) or recurring (R) | T | R |
b Read what Zara says.
The denominators of the fractions $\tfrac{1}{2}$, $\tfrac{1}{4}$ and $\tfrac{1}{8}$ are all powers of $2$.
The powers of $2$ are $2^1=2$, $2^2=4$, $2^3=8$, etc. There is a pattern in the equivalent decimals for $\tfrac{1}{2}$, $\tfrac{1}{4}$ and $\tfrac{1}{8}$: they are all terminating decimals. The decimals are $0.5$, $0.25$ and $0.125$. The pattern is 0.5, 0.25, 0.125. I think all unit fractions with a denominator that is a power of $2$ will be a terminating decimal that ends in $25$, apart from $\tfrac{1}{2^1}$ which just ends in $5$.
- Do you think Zara is correct? Test her idea on $\tfrac{1}{16}$ and $\tfrac{1}{32}$.
Explain your decisions. - What other patterns can you see in the table in Question 1? Test your ideas to see if they work.
- Discuss your ideas with other groups of learners in your class.
$2^4 = 16,\quad 2^5 = 32$
❓ EXERCISES
3. Here are five fraction cards.
a. Without doing any calculations, do you think these fractions are terminating or recurring decimals? Explain why.
b. Use a written method to convert the fractions to decimals.
c. Write the fractions in order of size, starting with the smallest.
👀 Show answer
a. All are terminating because each denominator’s prime factors are only $2$ and/or $5$ (after simplification): $\dfrac{8}{},\ \dfrac{4}{},\ \dfrac{10}{},\ \dfrac{20}{},\ \dfrac{5}{}$.
b. $A:\ \dfrac{5}{8}=0.625,\quad B:\ \dfrac{3}{4}=0.75,\quad C:\ \dfrac{7}{10}=0.7,\quad D:\ \dfrac{11}{20}=0.55,\quad E:\ \dfrac{3}{5}=0.6$
c. From smallest to largest (using decimals $0.55,\ 0.6,\ 0.625,\ 0.7,\ 0.75$): $D:\ \dfrac{11}{20},\ E:\ \dfrac{3}{5},\ A:\ \dfrac{5}{8},\ C:\ \dfrac{7}{10},\ B:\ \dfrac{3}{4}$.
4. Here are five fraction cards.
a. Without doing any calculations, do you think these fractions are terminating or recurring decimals? Explain why.
b. Use a written method to convert the fractions to decimals.
c. Write the fractions in order of size, starting with the smallest.
👀 Show answer
a. All are recurring because each denominator has a prime factor other than $2$ or $5$ (e.g., $3, 11$).
b. $A:\ \dfrac{5}{6}=0.8\overline{3},\quad B:\ \dfrac{2}{3}=0.\overline{6},\quad C:\ \dfrac{7}{12}=0.58\overline{3},\quad D:\ \dfrac{5}{9}=0.\overline{5},\quad E:\ \dfrac{3}{11}=0.\overline{27}$
c. Using decimals $0.\overline{27},\ 0.\overline{5},\ 0.58\overline{3},\ 0.\overline{6},\ 0.8\overline{3}$, the order (smallest first) is: $E:\ \dfrac{3}{11},\ D:\ \dfrac{5}{9},\ C:\ \dfrac{7}{12},\ B:\ \dfrac{2}{3},\ A:\ \dfrac{5}{6}$.
🧠 Think like a Mathematician
Task: Explore how calculators handle recurring decimals and fractions. Investigate what the S ⇔ D button does and how to match fractions to their decimal equivalents.
Scenario: A student converts four fractions to recurring decimals on a calculator. The calculator shows:
- 0.111111111
- 0.733333333
- 0.888888889
- 0.388888889
Questions:
👀 show answer
- a) The calculator display is rounded to a fixed number of digits. When a recurring decimal doesn’t fit exactly, the calculator ends with a 9 to show rounding (e.g., 0.888888889 is actually 0.888... recurring).
- b) - 0.111111111 → $\dfrac{1}{9}$ - 0.733333333 → $\dfrac{11}{15}$ - 0.888888889 → $\dfrac{8}{9}$ - 0.388888889 → $\dfrac{7}{18}$
- c) The S ⇔ D button toggles between fraction (standard form) and decimal (decimal form). Pressing it again switches back.
- d) - i. $\dfrac{7}{15} = 0.4666...$ - ii. $\dfrac{8}{11} = 0.7272...$
❓ EXERCISES
6. Use a calculator to convert these fractions to decimals.
a. $\dfrac{7}{9}$
b. $\dfrac{13}{20}$
c. $\dfrac{2}{15}$
d. $\dfrac{9}{40}$
👀 Show answer
a. $\dfrac{7}{9} = 0.\overline{7}$ (recurring)
b. $\dfrac{13}{20} = 0.65$ (terminating)
c. $\dfrac{2}{15} = 0.1\overline{3}$ (recurring)
d. $\dfrac{9}{40} = 0.225$ (terminating)
7. Marcus and Sofia are discussing the fraction $\dfrac{5}{13}$.
Marcus: My calculator tells me that $5 \div 13 = 0.38461538$, so I think that $\dfrac{5}{13}$ is a recurring decimal which I can write as $0.\overline{384615}$.
Sofia: I don’t think the calculator shows you enough decimal places to decide it is a recurring decimal.
👀 Show answer
$\dfrac{5}{13} = 0.\overline{384615}$, which is a recurring decimal with a 6-digit cycle. Marcus is correct, but Sofia’s caution is also valid because the calculator initially shows only a few digits. To be certain, more decimal places must be checked.
8. Use a calculator to convert these fractions to recurring decimals.
a. $\dfrac{2}{7}$ b. $\dfrac{9}{13}$ c. $\dfrac{11}{14}$
🔎 Reasoning Tip
Remember, when several digits repeat in the decimal, you only put a dot over the first and the last digit of the sequence that repeats, e.g.
$\dfrac{1}{7} = 0.\dot{1}4285\dot{7}$
👀 Show answer
a. $\dfrac{2}{7} = 0.\overline{285714}$
b. $\dfrac{9}{13} = 0.\overline{692307}$
c. $\dfrac{11}{14} = 0.78\overline{57}$
9. This is part of Kim’s homework.
Question: Write these fractions as decimals.
a. Use a calculator to check Kim’s homework.
b. Explain any mistakes she has made and write the correct answers.
🔎 Reasoning Tip
Change the improper fractions into mixed numbers first. Then use your answers to Question 1 to help.
👀 Show answer
i. $\dfrac{5}{12} = 0.41\overline{6}$ → Kim wrote $0.416$, but it should show as recurring.
ii. $\dfrac{10}{11} = 0.\overline{90}$ → Kim wrote $0.90$, but it is recurring, not terminating.
iii. $\dfrac{6}{7} = 0.\overline{857142}$ → Kim wrote $0.857142$, missing the recurring notation.
iv. $\dfrac{1}{37} = 0.\overline{027}$ → Kim wrote $0.027$, missing the recurrence.
10. Without using a calculator, write these fractions as decimals.
a. $\dfrac{4}{3}$ b. $\dfrac{13}{6}$ c. $\dfrac{19}{9}$ d. $\dfrac{45}{11}$
👀 Show answer
a. $\dfrac{4}{3} = 1.\overline{3}$
b. $\dfrac{13}{6} = 2.1\overline{6}$
c. $\dfrac{19}{9} = 2.\overline{1}$
d. $\dfrac{45}{11} = 4.\overline{09}$
11. This is part of Ada’s homework.
Use Ada’s method to write these lengths of time as recurring decimals.
a. 4 hours 20 minutes
b. 1 hour 40 minutes
c. 6 hours 10 minutes
d. 3 hours 50 minutes
👀 Show answer
a. $4 + \dfrac{20}{60} = 4 + \dfrac{1}{3} = 4.3\overline{3}$
b. $1 + \dfrac{40}{60} = 1 + \dfrac{2}{3} = 1.6\overline{6}$
c. $6 + \dfrac{10}{60} = 6 + \dfrac{1}{6} = 6.1\overline{6}$
d. $3 + \dfrac{50}{60} = 3 + \dfrac{5}{6} = 3.8\overline{3}$
12. Rajim has 8 weeks holiday a year.
There are 52 weeks in a year.
What fraction of the year does he have on holiday?
Write your answer as a decimal.
👀 Show answer
Fraction of year = $\dfrac{8}{52} = \dfrac{2}{13} = 0.\overline{153846}$
13. Sasha is told that $\dfrac{1}{15}=0.0\overline{6}$ and that $\dfrac{1}{22}=0.0\overline{45}$.
Without using a calculator, she must match each yellow fraction card with the correct blue decimal card.
Sasha thinks that $\dfrac{4}{15}=0.26$ and that $\dfrac{7}{22}=0.31\overline{8}$.
Do you think she is correct? Explain your answer.
👀 Show answer
$\dfrac{4}{15} = 4 \times 0.0\overline{6} = 0.26\overline{6}$, so rounding to 2dp gives $0.26$ — correct.
$\dfrac{7}{22} = 7 \times 0.0\overline{45} = 0.31\overline{81}$, which matches $0.318$ (recurring) — correct.
So, Sasha is correct in both cases.
🍬 Learning Bridge
You’ve just practised turning fractions into decimals using division and learned how to mark repeating parts with a bar or dots. Next, you’ll flip the idea: instead of calculating first, you’ll predict whether a fraction’s decimal will terminate or recur just by looking at its denominator.
- Key test: If the denominator’s prime factors are only 2s and/or 5s, the decimal terminates. Any other prime (3, 7, 11, …) means it recurs.
- Use patterns you’ve seen (like ninths and elevens) to spot repeating blocks quickly, and multiply known results (e.g. from
1/5or1/3) to get others like3/5or2/3. - Write answers clearly with recurring notation: bars (
0.\overline{71}) or dotted digits.
Quick check: Decide—terminating or recurring? 13/20 (only 2s & 5s → terminating) vs 5/12 (has a 3 → recurring).
You already know how to use equivalent fractions to convert a fraction to an equivalent decimal. For example: $\tfrac{1}{5}=\tfrac{2}{10}=0.2$
You also know how to use division to convert a fraction to an equivalent decimal. For example: $\tfrac{13}{25}=13 \div 25=0.52$
The decimals $0.2$ and $0.52$ are terminating decimals because they come to an end. You also know that you can write the fraction $\tfrac{1}{3}$ as $0.\dot{3}$ and that this is a recurring decimal because the digit $3$ is repeated forever. You can use what you already know about terminating and recurring decimals to deduce whether other fractions are terminating or recurring.
🔎 Reasoning Tip
You can use a written method or a calculator to use division to convert a fraction to an equivalent decimal.
❓ EXERCISES
14a. Copy and complete:
$\dfrac{1}{4}=0.25$ which is a terminating decimal
$\dfrac{2}{4}=2\times\dfrac{1}{4}=2\times 0.25=\;\;?$ which is a … decimal
$\dfrac{3}{4}=3\times\dfrac{1}{4}=3\times 0.25=\;\;?$ which is a … decimal
👀 Show answer
$\dfrac{2}{4}=0.5$ → terminating
$\dfrac{3}{4}=0.75$ → terminating
14b. Copy and complete:
$\dfrac{1}{5}=0.2$ which is a terminating decimal
$\dfrac{2}{5}=2\times\dfrac{1}{5}=2\times 0.2=\;\;?$ which is a … decimal
$\dfrac{4}{5}=4\times\dfrac{1}{5}=4\times 0.2=\;\;?$ which is a … decimal
👀 Show answer
$\dfrac{2}{5}=0.4$ → terminating
$\dfrac{4}{5}=0.8$ → terminating
15.
a. Work out the decimal equivalent of $\dfrac{1}{9}$.
b. Is $\dfrac{1}{9}$ a terminating or recurring decimal?
c. Write the decimal equivalents of these fractions. State whether each is terminating or recurring:
i. $\dfrac{2}{9}$ ii. $\dfrac{3}{9}$ iii. $\dfrac{4}{9}$ iv. $\dfrac{5}{9}$ v. $\dfrac{6}{9}$ vi. $\dfrac{7}{9}$ vii. $\dfrac{8}{9}$
👀 Show answer
$\dfrac{1}{9}=0.\overline{1}$ → recurring
i. $\dfrac{2}{9}=0.\overline{2}$ → recurring
ii. $\dfrac{3}{9}=\dfrac{1}{3}=0.\overline{3}$ → recurring
iii. $\dfrac{4}{9}=0.\overline{4}$ → recurring
iv. $\dfrac{5}{9}=0.\overline{5}$ → recurring
v. $\dfrac{6}{9}=\dfrac{2}{3}=0.\overline{6}$ → recurring
vi. $\dfrac{7}{9}=0.\overline{7}$ → recurring
vii. $\dfrac{8}{9}=0.\overline{8}$ → recurring
🧠 Think like a Mathematician
Task: Explore recurring decimals and their connection to fractions such as ninths. Reflect on statements about $\dfrac{9}{9}$ and $0.\overline{9}$.
Questions:
👀 show answer
- a) The recurring decimals 0.333… and 0.666… correspond to $\dfrac{1}{3}$ and $\dfrac{2}{3}$. These connect to $\dfrac{3}{9}$ and $\dfrac{6}{9}$, which simplify to the same values.
- b) Marcus says $\dfrac{9}{9} = 0.\overline{9}$. Zara says $\dfrac{9}{9} = 1$. Both are correct because $0.\overline{9}$ is another way of writing 1. They are not actually different answers, but two equal representations of the same value.
❓ EXERCISES
17.
a. Work out the decimal equivalent of $\dfrac{1}{8}$.
b. Is $\dfrac{1}{8}$ a terminating or recurring decimal?
c. Use your answers to parts a and b to write down the decimal equivalents of these fractions. Write if each decimal is terminating or recurring.
i. $\dfrac{2}{8}$ ii. $\dfrac{3}{8}$ iii. $\dfrac{4}{8}$ iv. $\dfrac{5}{8}$ v. $\dfrac{6}{8}$ vi. $\dfrac{7}{8}$
d. Look at your answers to part ci, iii and v. Did you write these fractions in their simplest form before you changed them into decimals? If you did, explain why. If you did not, do you think it would have been easier if you had?
👀 Show answer
a. $\dfrac{1}{8}=0.125$
b. Terminating.
c.
i. $\dfrac{2}{8}=\dfrac{1}{4}=0.25$ — terminating
ii. $\dfrac{3}{8}=0.375$ — terminating
iii. $\dfrac{4}{8}=\dfrac{1}{2}=0.5$ — terminating
iv. $\dfrac{5}{8}=0.625$ — terminating
v. $\dfrac{6}{8}=\dfrac{3}{4}=0.75$ — terminating
vi. $\dfrac{7}{8}=0.875$ — terminating
d. Simplifying first can make converting quicker: e.g., $\dfrac{2}{8}\to\dfrac{1}{4}$ and $\dfrac{6}{8}\to\dfrac{3}{4}$ are well‑known decimals ($0.25$, $0.75$). $\dfrac{3}{8}$ is already in simplest form, so we convert it directly to $0.375$.
🧠 Think like a Mathematician
Task: Investigate which fractions give recurring decimals, and try to find a general rule about denominators.
Questions:
👀 show answer
- a) Not all fractions with denominator 6 are recurring decimals. For example, $\dfrac{1}{2} = 0.5$ and $\dfrac{1}{3} = 0.\overline{3}$. Some terminate, some recur.
- b) Yes, all fractions with denominator 7 give recurring decimals because 7 is a prime number other than 2 or 5. None of the fractions simplify to have only factors 2 or 5 in the denominator.
- c) The difference is that 6 has factors 2 and 3. Denominators with factor 2 and/or 5 can produce terminating decimals, while denominators with other primes (like 7) always produce recurring decimals.
- d) - For denominator 11, all unit fractions give recurring decimals. - For denominator 12, some fractions terminate (e.g. $\dfrac{1}{4} = 0.25$) and some recur (e.g. $\dfrac{1}{3} = 0.\overline{3}$).
General rule: A fraction in simplest form gives a terminating decimal if and only if the denominator’s prime factors are only 2 or 5. Otherwise, the decimal is recurring.
❓ EXERCISES
19.
a. Here are five fraction cards.
Without doing any calculations, answer this question.
Are these fractions terminating or recurring decimals? Explain how you know.
b. All the numerators are changed from 1 to 2, so the cards now look like this:
Are these fractions terminating or recurring decimals? Explain your answer.
c. If all the numerators in part b are changed from 2 to 3, are the fractions terminating or recurring decimals? Explain your answer.
d. Look back at your answers to parts b and c and answer this question.
When a fraction has a denominator which has a factor of 3, is the fraction always equivalent to a recurring decimal?
Discuss your answer with other learners in your class.
👀 Show answer
a. All these fractions ($\tfrac{1}{3}, \tfrac{1}{6}, \tfrac{1}{9}, \tfrac{1}{12}, \tfrac{1}{15}$) are recurring decimals, because their denominators include prime factors other than only $2$ or $5$.
b. Changing the numerators to $2$ does not affect whether the decimals are terminating or recurring. They are still recurring decimals.
c. Changing the numerators to $3$ also does not affect the decimal type. They remain recurring decimals.
d. Yes — if the denominator has a factor of $3$ (or any prime other than $2$ or $5$), the fraction will produce a recurring decimal. This is because only denominators made from powers of $2$ and/or $5$ give terminating decimals.
20. Decide if these statements about proper fractions are ‘Always true’, ‘Sometimes true’ or ‘Never true’.
Justify your answers.
a. A fraction with a denominator of 7 is a recurring decimal.
b. A fraction with a denominator which is a multiple of 2 is a recurring decimal.
c. A fraction with a denominator which is a multiple of 10 is a terminating decimal.
d. A fraction with a denominator which is a power of 2 is a recurring decimal.
🔎 Reasoning Tip
The numbers which are powers of 2 are: $2^1 = 2$, $2^2 = 4$, $2^3 = 8$, etc.
👀 Show answer
a → Always true (since 7 has a prime factor other than 2 or 5).
b → Sometimes true (e.g. $\tfrac{1}{2}=0.5$ terminating, but $\tfrac{1}{6}=0.1\overline{6}$ recurring).
c → Sometimes true (e.g. $\tfrac{1}{10}$ terminating, but $\tfrac{1}{30}$ recurring because of factor 3).
d → Never true (e.g. $\tfrac{1}{8}=0.125$, always terminating).
21. Here are five fraction cards.
a. Without doing any calculations, answer this question. Do you think these fractions are terminating or recurring decimals? Explain why.
b. Which fraction is different from the others? Explain this difference. Does this change your answer to part a? Explain why.
c. Read what Sofia and Arun say.
Sofia: “Any fraction which has a denominator which is a multiple of 7 is a recurring decimal.”
Arun: “That’s not true, because $\tfrac{7}{14}=\tfrac{1}{2}$ which is not recurring and $\tfrac{7}{28}=\tfrac{1}{4}$ which is not recurring.”
What must Sofia add to her statement to make her statement true?
👀 Show answer
a. All the given fractions have denominators with a factor other than only 2 or 5, so most will be recurring decimals.
b. The fraction $\tfrac{3}{14}$ is different because it simplifies to $\tfrac{3}{14}$ (denominator 14 = $2\times7$). If simplified in certain cases (like multiples of 14 that reduce to denominator 2), it could give a terminating decimal. This shows some exceptions exist.
c. Sofia should add: “…provided the denominator, when simplified, still contains a factor of 7.” This makes her statement correct, since if the factor of 7 cancels out, the decimal can be terminating.
22. This is part of Ed’s homework.
🔎 Reasoning Tip
Think about the number of minutes in an hour.
Use Ed’s method to decide if these lengths of time, as a fraction of an hour, are terminating or recurring decimals.
a. 20 minutes
b. 36 minutes
c. 45 minutes
d. 55 minutes
e. 1 hour 8 minutes
f. 3 hours 21 minutes
👀 Show answer
a. $20/60=1/3=0.\overline{3}$ → recurring
b. $36/60=3/5=0.6$ → terminating
c. $45/60=3/4=0.75$ → terminating
d. $55/60=11/12=0.91\overline{6}$ → recurring
e. $68/60=34/30=17/15=1.1\overline{3}$ → recurring
f. $(201/60)=67/20=3.35$ → terminating
23. Without using a calculator, decide if these fractions are terminating or recurring decimals.
a. $\dfrac{8}{3}=2.\overline{6}$
b. $\dfrac{21}{5}=4.2$
c. $\dfrac{28}{9}=3.\overline{1}$
d. $\dfrac{39}{12}=\dfrac{13}{4}=3.25$
🔎 Reasoning Tip
Change the improper fractions into mixed numbers first.
👀 Show answer
a → recurring
b → terminating
c → recurring
d → terminating
24. The table shows the number of hours that six friends work each week.
There are 168 hours in one week.
| Abi | Bim | Caz | Dave | Enid | Fin |
|---|---|---|---|---|---|
| 21 | 28 | 32 | 35 | 40 | 42 |
a. Sort the friends into two groups according to the number of hours they work, as a fraction of one week. Explain the criteria you used to group them.
b. Sort the friends into two different groups according to the number of hours they work, as a fraction of one week. Explain the criteria you used to group them this time.
👀 Show answer
a. As fractions of 168 hours:
- Abi: $21/168=1/8$ → terminating
- Bim: $28/168=1/6$ → recurring
- Caz: $32/168=2/21$ → recurring
- Dave: $35/168=5/24$ → recurring
- Enid: $40/168=10/42=5/21$ → recurring
- Fin: $42/168=1/4$ → terminating
Group 1 (terminating): Abi, Fin
Group 2 (recurring): Bim, Caz, Dave, Enid
b. Another way: group them by whether their hours divide evenly into 168.
Divisors: 21, 28, 42 divide 168 exactly → Abi, Bim, Fin.
Non-divisors: 32, 35, 40 → Caz, Dave, Enid.
⚠️ Be careful! Terminating vs Recurring Decimals
- Do not stop division too early. Some fractions look like they end, but the digits start repeating later (e.g. $1 \div 7 = 0.142857\ldots$).
- Recurring decimals are not approximations. Write them with a bar or dots to show infinite repetition: $0.\overline{71}$ or $0.\dot{7}\dot{1}$, not $0.71$.
- Check the denominator’s factors. Only denominators with prime factors 2 and/or 5 give terminating decimals (e.g. $\tfrac{3}{40}=0.075$). Any other prime factor (3, 7, 11, …) means recurring.
- Be careful with calculators. Some show $0.888888889$ instead of $0.\overline{8}$ — the 9 at the end is just rounding, not part of the repeat.
- Improper fractions still follow the same rule. $\tfrac{19}{9}=2.\overline{1}$, not $2.1$.
- Equivalent fractions matter. Simplify first: $\tfrac{6}{12}=\tfrac{1}{2}=0.5$ (terminating), not recurring.