Lesson 7.5 · 30 min

Recurring Decimals to Fractions — the Algebraic Method

Converting any recurring decimal to a fraction by multiplying by 10, 100 or 1000 and subtracting — including decimals where only part recurs — and writing it up as a proof.

In this lesson you will learn to…

Convert a recurring decimal such as $0. \dot{3} \dot{6}$ to a fraction using algebra
Choose the right power of 10 from the length of the repeating block
Convert decimals where only part recurs, such as $0.1 \dot{6}$ and $2.0 \dot{4} \dot{5}$
Show that a recurring decimal equals a given fraction, in full exam-style working

Why multiplying and subtracting works

Every recurring decimal is a fraction, and algebra finds it. The trick is to make two numbers with exactly the same infinite tail, then subtract so the tail disappears.

$\begin{aligned} x &= 0.4444\ldots \\ 10x &= 4.4444\ldots \\ 10x - x = 9x &= 4 \quad\Rightarrow\quad x = \tfrac{4}{9} \end{aligned}$

Longer repeating blocks

If two digits repeat, multiply by $100$ ; if three, by $1000$ . The tails only line up when you shift by a whole block.

When only part of the decimal recurs

In $0.1\dot{6}$ the 1 does not repeat. Multiply first to move it in front of the decimal point, then multiply again by the block length. Subtract those two — both have tail $.666\ldots$

Writing it as a proof

“Show that $0.\dot{2}\dot{7} = \frac{3}{11}$ ” needs every step. Converting $\frac{3}{11}$ to a decimal on a calculator does not count.

Key rules and formulas

The method
1. Let $x$ equal the decimal, written out with several repeats. 2. Multiply by $1 0^{n}$ , where $n$ is the length of the repeating block. 3. Subtract so the recurring tails cancel. 4. Solve for $x$ and simplify.
When only part recurs
First multiply so the non-recurring digits are before the point (e.g. $10 x$ for $0.1 \dot{6}$ ), then by $1 0^{n}$ more (e.g. $100 x$ ). Subtract those two.
A shortcut to check
$0. \overset{a}{˙} \dot{b} = \frac{ab}{99} 0. \overset{a}{˙} b \overset{c}{˙} = \frac{ab c}{999}$
Useful for checking, but exam questions that say "show that" or "prove" need the algebra.

Worked examples

Worked example 1

A single recurring digit

Write

0. \dot{5}

as a fraction.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

A three-digit block

Write

0. \dot{1} 0 \dot{8}

as a fraction in its simplest form.

Try it yourself first, then reveal the solution one step at a time.

Worked example 3

Part recurring, bigger than 1

Write

2.0 \dot{4} \dot{5}

as a mixed number.

Try it yourself first, then reveal the solution one step at a time.

Practice: recurring decimals to fractions

11 questions

Give fractions in their simplest form, e.g. 4/11 or 2 1/22.

Question 1 of 11Type the answer
Write $0. \dot{7}$ as a fraction.
Your answer
Question 2 of 11Type the answer
Write $0. \dot{1} \dot{8}$ as a fraction in its simplest form.
Your answer
Question 3 of 11Type the answer
Write $0. \dot{2} 1 \dot{6}$ as a fraction in its simplest form.
Your answer
Question 4 of 11Type the answer
Write $0.8 \dot{3}$ as a fraction in its simplest form.
Your answer
Question 5 of 11Type the answerChallenge
Write $0.4 \dot{1} \dot{2}$ as a fraction in its simplest form.
Your answer
Question 6 of 11Type the answer
Write $1. \dot{3} \dot{6}$ as a mixed number.
Your answer
Question 7 of 11Type the answerChallenge
Write $3.2 \dot{7}$ as an improper fraction in its simplest form.
Your answer
Question 8 of 11Multiple choice
To convert $x = 0.0 \dot{3} \dot{7}$ to a fraction, which two equations should you subtract?
Question 9 of 11Type the answerChallenge
Work out $0. \dot{4} + 0. \dot{2} \dot{7}$ . Give your answer as a fraction in its simplest form.
Your answer
Question 10 of 11Fill the gaps
Complete the proof that $0.2 \dot{5} = \frac{23}{90}$ .
Let $x = 0.2555 \dots$ . Then $10 x =$ Gap a and $100 x =$ Gap b. Subtracting, $90 x =$ Gap c, so $x = \frac{23}{90}$ .
Question 11 of 11True or false
$0. \dot{1} \dot{2} = 0.1 \dot{2}$

In this lesson you will learn to…

Why multiplying and subtracting works

Longer repeating blocks

When only part of the decimal recurs

Writing it as a proof

Key rules and formulas

The method

When only part recurs

A shortcut to check

Worked examples

Practice: recurring decimals to fractions