Fractions & recurring decimals

Start here: two kinds of decimal

Type $1 \div 4$ into a calculator. You get $0.25$ — it stops. Now type $1 \div 3$ . You get $0.33333\ldots$ — it never stops.

Every fraction turns into one of these two kinds of decimal:

a terminating decimal (ends neatly), or
a recurring decimal (a group of digits repeats forever).

The interesting question: can you tell which kind, just by looking at the bottom of the fraction? Yes — this topic shows how.

What you will learn

convert a fraction to a decimal by long division,
recognise a terminating decimal from a recurring (repeating) decimal,
predict from the denominator’s prime factors whether the decimal will terminate,
use recurring-decimal notation (the bar over repeating digits),
convert a recurring decimal back to a fraction using simple algebra.

1. Fraction to decimal

To convert $\dfrac{a}{b}$ to a decimal, divide $a$ by $b$ . The result is either a terminating decimal (ends) or a recurring decimal (has a block of digits that repeats forever).

Worked example E Very easy: a half

$\dfrac{1}{2} = 1 \div 2 = 0.5$ . Terminates after one decimal place.

Why? Because $\dfrac{1}{2} = \dfrac{5}{10}$ , and dividing by $10$ always gives a tidy one-digit decimal.

Worked example E Very easy: a quarter

$\dfrac{1}{4} = 1 \div 4 = 0.25$ . It terminates — the division ends with no remainder.

Worked example 1 Terminating

$\dfrac{7}{8} = 7 \div 8$ . Long-dividing: $7.000 \div 8 = 0.875$ . This terminates after three decimal places.

Worked example 2 Recurring

$\dfrac{1}{3} = 1 \div 3 = 0.3333\ldots$ The digit $3$ repeats forever. Write it compactly as $0.\overline{3}$ (a bar over the repeating block).

Other examples:

$\dfrac{1}{6} = 0.1\overline{6}$ (digit $6$ repeats after a non-repeating start).
$\dfrac{1}{7} = 0.\overline{142857}$ (six-digit block $142857$ repeats).

2. Predicting from the denominator

Write the denominator in simplest form as a product of primes. If those primes are only $2$ and/or $5$ , the decimal terminates. If any other prime appears, it recurs.

Terminating or recurring?

Rule

Simplify the fraction first. Let $b$ be the denominator. Factorise $b$ into primes.

Terminates if the only prime factors of $b$ are $2$ and/or $5$ .
Recurs otherwise.

Why this works

Decimals are powers of $10 = 2 \times 5$ . A fraction terminates only when its denominator divides some power of $10$ , which requires only the primes $2$ and $5$ .

Worked example 3 Predict, then verify

Will $\dfrac{7}{20}$ terminate? $20 = 2^2 \times 5$ . Only primes $2$ and $5$ . Yes - terminates.

Check: $\dfrac{7}{20} = 0.35$ . Terminates after two decimal places. ✓

Will $\dfrac{5}{12}$ terminate? $12 = 2^2 \times 3$ . The prime $3$ appears. Recurs.

Check: $\dfrac{5}{12} = 0.41\overline{6}$ . Recurs. ✓

3. Recurring decimal to fraction

Use a bit of algebra: call the decimal $x$ , multiply by a power of $10$ that shifts the repeating block one full length, subtract.

Worked example E Very easy: the 10x trick on 0.111...

Let $x = 0.\overline{1} = 0.1111\ldots$

Multiply by $10$ : $10x = 1.1111\ldots$
Subtract: $10x - x = 1$ , so $9x = 1$ .
$x = \dfrac{1}{9}$ .

This is the simplest case of the “multiply-then-subtract” method. Every single-digit recurring decimal works the same way.

Worked example 4 Convert 0.7777 recurring to a fraction

Let $x = 0.\overline{7}$ .

Multiply by $10$ (one full block): $10 x = 7.\overline{7}$ .
Subtract the first equation: $10 x - x = 7.\overline{7} - 0.\overline{7} = 7$ .
$9 x = 7$ , so $x = \dfrac{7}{9}$ .

Check: $7 \div 9 = 0.7777\ldots$ ✓

Worked example 5 Two-digit recurring block

Let $x = 0.\overline{27} = 0.272727\ldots$ .

The block is two digits, so multiply by $100$ : $100 x = 27.\overline{27}$ .
Subtract: $99 x = 27$ .
$x = \dfrac{27}{99} = \dfrac{3}{11}$ .

Check: $3 \div 11 = 0.272727\ldots$ ✓

Practice: Year 8 core

Fluency

Fraction to decimal

Write $\dfrac{1}{2}$ as a decimal.
Does $\dfrac{1}{5}$ terminate or recur? Write its decimal.
Write $\dfrac{1}{10}$ as a decimal.
Write $\dfrac{3}{4}$ as a decimal.
Write $\dfrac{3}{8}$ as a decimal.
Write $\dfrac{11}{20}$ as a decimal.
Write $\dfrac{2}{3}$ as a decimal (use the recurring bar).
Write $\dfrac{4}{9}$ as a decimal.
Write $\dfrac{7}{11}$ as a decimal (use the bar over the repeating block).
Write $\dfrac{5}{6}$ as a decimal.

Fluency

Predict terminating or recurring

Will $\dfrac{9}{40}$ terminate? Justify.
Will $\dfrac{7}{15}$ terminate? Justify.
Will $\dfrac{13}{50}$ terminate? Justify.
Will $\dfrac{5}{24}$ terminate? Justify.
Will $\dfrac{17}{125}$ terminate? Justify.

Fluency

Recurring to fraction

Convert $0.\overline{5}$ to a fraction.
Convert $0.\overline{8}$ to a fraction.
Convert $0.\overline{12}$ to a fraction.
Convert $0.\overline{45}$ to a fraction (and simplify).
Convert $0.\overline{123}$ to a fraction (and simplify).

Reasoning

Explain and spot the mistake

Dev says " $\dfrac{1}{9} = 0.1$ ". Is Dev correct? If not, what is the correct decimal?
Explain why $\dfrac{1}{6}$ recurs but $\dfrac{1}{5}$ terminates.
Kim writes $0.\overline{9} = 0.999\ldots$ and claims this is less than $1$ . Is Kim correct? (Hint: try the algebra trick from Worked example 4.)
Show that $\dfrac{3}{25}$ terminates by writing $25$ as a power of primes.

Problem solving

Applications

A carpenter needs to split a $1$ m length into $7$ equal parts. Is each part’s length a terminating or recurring decimal in metres? Explain.
Write $\dfrac{1}{7}$ as a decimal. What is the smallest integer $n$ such that $7 \mid 10^n - 1$ ? (This is the length of the repeating block.)
Convert $0.\overline{142857}$ back to a fraction. (You should recognise the answer.)
A recipe says “use $\dfrac{1}{3}$ cup of butter”. A measuring cup has decimal markings. Explain how the cook should round.

Challenge

Reasoning

Harder reasoning

Convert $0.1\overline{6}$ to a fraction. (Hint: note the non-repeating “1” at the start; multiply by a power of $10$ to clear the non-repeating part first.)
Convert $0.4\overline{72}$ to a fraction.
Two fractions have the same decimal expansion $0.\overline{45}$ . Are they equal? Explain.
Without computing, decide whether $\dfrac{1}{2^3 \times 5 \times 7}$ terminates. Justify.