Sets of Numbers

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Senior 1 · Mathematics

Sets of Numbers

5 lessons

Lesson 4 of 5

Converting between decimals and fractions

24 min

Decimals that stop and decimals that repeat

Type $1 \div 4$ into a calculator and it shows 0.25: the decimal stops. Type $1 \div 3$ and it shows 0.3333333333, filling the screen with 3s because the decimal never stops. Every fraction behaves in one of these two ways, and there is a rule that tells you which, without dividing.

Definition

Terminating decimal: a decimal with a finite number of digits after the decimal point, such as $0.25$ or $3.125$ . These are the decimal numbers, the elements of $\mathbb{D}$ .

Recurring (repeating) decimal: a decimal in which one digit or a block of digits repeats forever, such as $0.333\ldots$ or $0.142857142857\ldots$ . The repeating block is called the period.

Definition

Notation for recurring decimals: put a dot over the first and last digits of the repeating block, or a bar over the whole block.

Decimal	Dot notation	Bar notation	Period
$0.333\ldots$	$0.\dot{3}$	$0.\overline{3}$	3
$0.1666\ldots$	$0.1\dot{6}$	$0.1\overline{6}$	6
$0.363636\ldots$	$0.\dot{3}\dot{6}$	$0.\overline{36}$	36
$0.142857142857\ldots$	$0.\dot{1}4285\dot{7}$	$0.\overline{142857}$	142857

From a fraction to a decimal

Divide the numerator by the denominator. Watch the remainders: they explain everything. Here is $1 \div 7$ , written as $1.000000\ldots \div 7$ :

Step	Divide	Digit of the answer	Remainder
1	$10 \div 7$	1	3
2	$30 \div 7$	4	2
3	$20 \div 7$	2	6
4	$60 \div 7$	8	4
5	$40 \div 7$	5	5
6	$50 \div 7$	7	1

After step 6 the remainder is 1 again, which is where we started. So the same six steps repeat for ever:

Dividing 1 by 7: the remainders go round a cycle of six, producing the digits 1, 4, 2, 8, 5, 7 again and again.

\frac{1}{7} = 0.142857142857\ldots = 0.\dot{1}4285\dot{7}.

Important

Every fraction gives either a terminating decimal or a recurring decimal

Why: when you divide by $b$ , every remainder is one of $0, 1, 2, \ldots, b - 1$ . If a remainder of 0 appears, the division stops and the decimal terminates. If 0 never appears, there are only $b - 1$ other possible remainders, so within $b$ steps some remainder must come back. From then on the division repeats itself, and so do the digits. That is why the period of $\frac{1}{7}$ is at most 6 digits long.

Important

Rule: a fraction in its simplest form gives a terminating decimal exactly when its denominator has no prime factors other than 2 and 5.

Why: a terminating decimal is a fraction $\frac{m}{10^k}$ , and $10^k = 2^k \times 5^k$ . So a denominator made only of 2s and 5s can be built up to a power of 10, for example $\frac{7}{40} = \frac{7}{2^3 \times 5} = \frac{7 \times 25}{2^3 \times 5^3} = \frac{175}{1000}$ . A prime factor such as 3 or 7 can never be cancelled into a power of 10.

Worked example 1: terminating or recurring?

Without dividing, decide whether each fraction gives a terminating or a recurring decimal, then check by dividing: (a) $\frac{7}{40}$ (b) $\frac{5}{12}$ (c) $\frac{9}{15}$

Solution

$40 = 2^3 \times 5$ : only 2s and 5s, so it terminates. $\frac{7}{40} = \frac{175}{1000} = 0.175$ .
$12 = 2^2 \times 3$ : the factor 3 means it recurs. $\frac{5}{12} = 0.41666\ldots = 0.41\dot{6}$ .
First simplify: $\frac{9}{15} = \frac{3}{5}$ . The denominator 5 means it terminates: $\frac{3}{5} = 0.6$ . (The 3 in 15 cancelled, so it did not matter.)

From a decimal to a fraction

A terminating decimal is already a fraction over a power of 10: write the digits over 10, 100, 1000, … and simplify. A recurring decimal needs a trick: multiply it by a power of 10 that shifts one full period, then subtract, so that the endless tails cancel.

Worked example 2: terminating decimals

Write (a) 0.375 and (b) 2.35 as fractions in simplest form.

Solution

\text{(a)}\quad 0.375 = \frac{375}{1000} = \frac{375 \div 125}{1000 \div 125} = \frac{3}{8}

\text{(b)}\quad 2.35 = \frac{235}{100} = \frac{47}{20} = 2\frac{7}{20}

Worked example 3: a pure recurring decimal

Write $0.\dot{3}\dot{6} = 0.363636\ldots$ as a fraction in simplest form.

Solution The period, 36, has 2 digits, so multiply by $10^{2} = 100$ .

\begin{aligned} x &= 0.363636\ldots \\ 100x &= 36.363636\ldots \\ 100x - x &= 36 \\ 99x &= 36 \\ x &= \frac{36}{99} = \frac{4}{11} \end{aligned}

Check: $4 \div 11 = 0.3636\ldots$ ✓

Worked example 4: a decimal that starts repeating later

Write $0.1\dot{6} = 0.1666\ldots$ as a fraction.

Solution First move the non-repeating digit to the left of the point, then shift one period further, and subtract.

\begin{aligned} 10x &= 1.666\ldots \\ 100x &= 16.666\ldots \\ 100x - 10x &= 15 \\ 90x &= 15 \\ x &= \frac{15}{90} = \frac{1}{6} \end{aligned}

Tip

Shortcut for a decimal that repeats straight after the point: write the period over as many 9s as it has digits.

0.\dot{7} = \frac{7}{9}, \qquad 0.\dot{3}\dot{6} = \frac{36}{99} = \frac{4}{11}, \qquad 0.\dot{1}2\dot{3} = \frac{123}{999} = \frac{41}{333}.

The 9s come from the subtraction step: $10^{k} - 1$ is a row of $k$ nines.

Fractions worth knowing by heart

Fraction	Decimal	Type
$\frac{1}{2}$	$0.5$	terminating
$\frac{1}{3}$	$0.\dot{3}$	recurring
$\frac{1}{4}$	$0.25$	terminating
$\frac{1}{5}$	$0.2$	terminating
$\frac{1}{6}$	$0.1\dot{6}$	recurring
$\frac{1}{8}$	$0.125$	terminating
$\frac{1}{9}$	$0.\dot{1}$	recurring
$\frac{1}{11}$	$0.\dot{0}\dot{9}$	recurring

Check each one against the rule about 2s and 5s: the denominators 2, 4, 5 and 8 are built from 2s and 5s; 3, 6, 9 and 11 are not.

Warning

A rounded decimal is not the fraction. $0.33 \neq \frac{1}{3}$ and $0.67 \neq \frac{2}{3}$ ; they are approximations. Keep the fraction, or the recurring decimal, when you need the exact value.
Do not put the period over 100. $0.\dot{3}\dot{6} \neq \frac{36}{100}$ , because $\frac{36}{100} = 0.36$ exactly, which stops. The denominator for a two-digit period is 99.

Important

A surprise: $0.\dot{9} = 1$ . Let $x = 0.999\ldots$ . Then $10x = 9.999\ldots$ , so $10 x - x = 9$ , $9 x = 9$ and $x = 1$ . The two decimals are two ways of writing the same number: there is no number between them.

Khan Academy: Converting repeating decimals to fractions 1 (4 min). Sal uses the same multiply-and-subtract method as Worked example 3. Pause before each step and predict what he will multiply by.

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Practice

Do questions 1 to 3 without a calculator. The last two are harder.

Write as decimals: (a) $\frac{3}{8}$ (b) $\frac{7}{20}$ (c) $\frac{2}{9}$ (d) $\frac{5}{6}$
Without dividing, say which of these give terminating decimals: $\frac{9}{16}$ , $\frac{7}{30}$ , $\frac{11}{125}$ , $\frac{13}{24}$ .
Write as fractions in simplest form: (a) 0.45 (b) 1.125 (c) 0.08
Write as fractions in simplest form: (a) $0.\dot{7}$ (b) $0.\dot{4}\dot{5}$ (c) $0.\dot{1}2\dot{3}$
Write as fractions in simplest form: (a) $0.8\dot{3}$ (b) $1.2\dot{7}$
Find the period of the decimal for $\frac{1}{13}$ , and say how many digits it has.
Explain, using remainders, why the decimal for $\frac{3}{17}$ must start repeating within 16 digits.

Show answersHide answers

(a) $0.375$ (b) $0.35$ (c) $0.\dot{2}$ (d) $0.8\dot{3}$
$\frac{9}{16}$ ( $16 = 2^{4}$ ) and $\frac{11}{125}$ ( $125 = 5^{3}$ ) terminate. $\frac{7}{30}$ (factor 3) and $\frac{13}{24}$ (factor 3) recur.
(a) $\frac{45}{100} = \frac{9}{20}$ (b) $\frac{1125}{1000} = \frac{9}{8}$ (c) $\frac{8}{100} = \frac{2}{25}$
(a) $\frac{7}{9}$ (b) $\frac{45}{99} = \frac{5}{11}$ (c) $\frac{123}{999} = \frac{41}{333}$
(a) $10x = 8.333\ldots$ , $100x = 83.333\ldots$ , $90 x = 75$ , $x = \frac{75}{90} = \frac{5}{6}$ (b) $10x = 12.777\ldots$ , $100x = 127.777\ldots$ , $90 x = 115$ , $x = \frac{115}{90} = \frac{23}{18}$
$\frac{1}{13} = 0.076923076923\ldots = 0.\dot{0}7692\dot{3}$ : the period 076923 has 6 digits.
Dividing by 17, the only possible non-zero remainders are 1 to 16. The remainder is never 0 (17 is not made of 2s and 5s), so within 16 steps a remainder must repeat, and from then on the digits repeat. (In fact the period of $\frac{3}{17}$ has exactly 16 digits.)

Sources and credits

5.5: Decimals and Fractions (Part 1), from Prealgebra by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
3.4: Rational Numbers, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
The remainder argument and the rule about the prime factors 2 and 5. Written by Ganzaa.
Video: Converting repeating decimals to fractions 1, Khan Academy, embedded from YouTube.

The Rwandan worked examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1Decimals that stop and decimals that repeat

2From a fraction to a decimal

3From a decimal to a fraction

4Fractions worth knowing by heart

Decimals that stop and decimals that repeat

From a fraction to a decimal

From a decimal to a fraction

Fractions worth knowing by heart