Recurring Decimals to Fractions

What you'll learn

How to read recurring decimal notation, including dots over repeating digits.
How fractions can become recurring decimals.
How to prove algebraically that a recurring decimal equals a fraction.
How to handle whole numbers and calculations involving recurring decimals.

1. Fractions and decimals

A fraction is another way of writing a division. For example, $711\frac{7}{11}$ means 7 divided by 11.

The denominator is the bottom number in a fraction. It tells you what you are dividing by.

Some fractions give decimals that stop. Others give decimals that repeat forever.

Definition

Terminating and recurring decimals

A terminating decimal stops after a fixed number of decimal places, such as 0.25.
A recurring decimal has a digit or block of digits that repeats forever, such as $0.7˙0.\dot{7}$ .

Recurring notation

A dot above one digit means that digit repeats forever:

0.7˙=0.777…0.\dot{7}=0.777\ldots

Two dots show the first and last digit in the repeating block:

0.3˙6˙=0.363636…0.\dot{3}\dot{6}=0.363636\ldots

and

0.24˙9˙=0.2494949…0.2\dot{4}\dot{9}=0.2494949\ldots

Here, the 2 does not repeat; only the block 49 repeats.

Example

Convert $\frac{7}{11}$ to a decimal

Think of the fraction as a division: 7 divided by 11.
11 does not go into 7, so start with 0 point and divide 70 by 11. This gives 6 with remainder 4.
Now divide 40 by 11. This gives 3 with remainder 7.
The remainder 7 has appeared again, so the same digits will repeat. The answer is:
$711=0.636363…=0.6˙3˙\frac{7}{11}=0.636363\ldots=0.\dot{6}\dot{3}$

Tip

Spot the repeat

In long division, when the same remainder appears again, the decimal digits from that point will repeat forever.

2. The algebra trick

To turn a recurring decimal into a fraction, we usually use algebra.

The main idea is to call the decimal $x$ , multiply by a power of 10, then subtract so the repeating part cancels.

A power of 10 means numbers such as 10, 100, 1000, and so on. Multiplying by 10 moves every digit one place to the left; multiplying by 100 moves every digit two places to the left.

The picture below shows why subtracting works: once the recurring tails match, the infinite repeating part disappears.

Diagram showing recurring decimal tails lining up before subtraction

Key Idea

The cancellation trick

Choose two multiples of your decimal so that their recurring tails are identical. When you subtract, the repeating digits cancel, leaving a normal equation to solve.

3. One repeating digit straight after the decimal point

A pure recurring decimal starts repeating immediately after the decimal point.

For example, $0.6˙0.\dot{6}$ means 0.666...

Example

Prove algebraically that $0. \dot{6} = \frac{2}{3}$

Let $x$ be the recurring decimal.
$x=0.666…x=0.666\ldots$
Multiply by 10 so the repeating digits line up.
$10x=6.666…10x=6.666\ldots$
Subtract the original equation from the new equation.
$10x−x=6.666…−0.666…9x=6\begin{aligned} 10x-x&=6.666\ldots-0.666\ldots\\ 9x&=6 \end{aligned}$
Divide by 9 and simplify.
$x=69=23x=\frac{6}{9}=\frac{2}{3}$

A fraction is in simplest form when the top and bottom numbers have no common factor other than 1. A common factor is a number that divides both exactly.

4. A non-repeating digit before the repeat

Some decimals have a digit after the decimal point that does not repeat, followed by a digit that does repeat.

For example, $0.38˙=0.3888…0.3\dot{8}=0.3888\ldots$

Here, the 3 is non-recurring, and the 8 recurs.

Tip

Which powers of 10?

If there are $n$ non-recurring decimal digits and the recurring block has length $r$ , compare $10^{n+r}x$ with $10^n x$ . This lines up the recurring tails.

Example

Write $0.3 \dot{8}$ as a fraction in its simplest form

Let $x$ be the decimal.
$x=0.3888…x=0.3888\ldots$
The recurring part starts after one decimal digit, so first multiply by 10.
$10x=3.888…10x=3.888\ldots$
One full recurring digit later, multiply by 100.
$100x=38.888…100x=38.888\ldots$
Subtract the two equations so the recurring 8s cancel.
$100x−10x=38.888…−3.888…90x=35\begin{aligned} 100x-10x&=38.888\ldots-3.888\ldots\\ 90x&=35 \end{aligned}$
Solve for $x$ and simplify.
$x=3590=718x=\frac{35}{90}=\frac{7}{18}$

Common Mistake

Multiplying by the wrong amount

For a decimal like $0.38˙0.3\dot{8}$ , do not just multiply by 10 and stop. The clean exam method is to compare $100 x$ with $10 x$ , because both have the same recurring tail.

5. Repeating blocks with more than one digit

If the repeating block has 2 digits, multiply by 100. If it has 3 digits, multiply by 1000.

Example

Write $0. \dot{4} \dot{7}$ as a fraction

Let $x$ be the decimal.
$x=0.474747…x=0.474747\ldots$
The repeating block has 2 digits, so multiply by 100.
$100x=47.474747…100x=47.474747\ldots$
Subtract the original equation.
$100x−x=47.474747…−0.474747…99x=47\begin{aligned} 100x-x&=47.474747\ldots-0.474747\ldots\\ 99x&=47 \end{aligned}$
Divide by 99.
$x=4799x=\frac{47}{99}$

6. Whole numbers before the decimal point

The same method works when there is a whole number before the decimal point. Your final answer may be an improper fraction, where the top number is larger than the bottom number. That is completely fine.

Example

Write $3.2 \dot{6} \dot{3}$ as a fraction in its simplest form

Write the decimal out so the pattern is clear.
$3.26˙3˙=3.263636…3.2\dot{6}\dot{3}=3.263636\ldots$
Let $x$ equal the decimal.
$x=3.263636…x=3.263636\ldots$
The non-recurring digit is 2, and the recurring block is 63. So compare $1000 x$ with $10 x$ .
$1000x=3263.636363…10x=32.636363…\begin{aligned} 1000x&=3263.636363\ldots\\ 10x&=32.636363\ldots \end{aligned}$
Subtract.
$1000x−10x=3263.636363…−32.636363…990x=3231\begin{aligned} 1000x-10x&=3263.636363\ldots-32.636363\ldots\\ 990x&=3231 \end{aligned}$
Divide by 990 and simplify.
$x=3231990=359110x=\frac{3231}{990}=\frac{359}{110}$

Tip

Keep the whole number inside $x$

You do not need to split off the whole number first. Let $x$ equal the entire decimal and use the same subtraction method.

7. Using a letter for a digit

Sometimes a question uses a letter to represent a digit.

An integer is a whole number, such as 0, 1, 2, or 9. If a digit $a$ is between 1 and 9, then it can stand for any one of those whole-number digits.

Example

Prove that $0. \dot{0} \overset{a}{˙} = \frac{a}{99}$ for a digit $a$

The dots show that the block 0a repeats forever.
$x=0.0a0a0a…x=0.0a0a0a\ldots$
The repeating block has 2 digits, so multiply by 100.
$100x=a.0a0a…100x=a.0a0a\ldots$
Subtract the original decimal.
$100x−x=a.0a0a…−0.0a0a…99x=a\begin{aligned} 100x-x&=a.0a0a\ldots-0.0a0a\ldots\\ 99x&=a \end{aligned}$
Divide by 99.
$x=a99x=\frac{a}{99}$

8. Calculations with recurring decimals

For multiplication or division questions, do not try to multiply the recurring decimals directly. Convert each recurring decimal into a fraction first.

Example

Work out $0.4 \dot{2} \div 0. \dot{3}$

First convert $0.42˙0.4\dot{2}$ into a fraction.
$x=0.4222…100x=42.222…10x=4.222…90x=38x=3890=1945\begin{aligned} x&=0.4222\ldots\\ 100x&=42.222\ldots\\ 10x&=4.222\ldots\\ 90x&=38\\ x&=\frac{38}{90}=\frac{19}{45} \end{aligned}$
Convert $0.3˙0.\dot{3}$ into a fraction.
$0.3˙=130.\dot{3}=\frac{1}{3}$
Replace the recurring decimals with fractions.
$0.42˙÷0.3˙=1945÷130.4\dot{2}\div 0.\dot{3}=\frac{19}{45}\div\frac{1}{3}$
Dividing by a fraction means multiplying by its reciprocal.
$1945÷13=1945×31=1915\frac{19}{45}\div\frac{1}{3}=\frac{19}{45}\times\frac{3}{1}=\frac{19}{15}$

Exam technique

In the exam

Write the recurring decimal out with a few extra digits so you can see exactly what repeats.
Choose powers of 10 that make the recurring tails match, then subtract the equations.
Always simplify your final fraction; for calculations, convert every recurring decimal before multiplying or dividing.

Self review

Check yourself

Can you explain why $100 x$ and $10 x$ are compared for a decimal like $0.74˙0.7\dot{4}$ ?
Can you convert $0.5˙0.\dot{5}$ into a fraction using algebra?
Can you work out $0.21˙×0.3˙0.2\dot{1}\times 0.\dot{3}$ by converting both decimals first?

Recurring Decimals to Fractions

What you'll learn

How to read recurring decimal notation, including dots over repeating digits.
How fractions can become recurring decimals.
How to prove algebraically that a recurring decimal equals a fraction.
How to handle whole numbers and calculations involving recurring decimals.

1. Fractions and decimals

A fraction is another way of writing a division. For example, $711\frac{7}{11}$ means 7 divided by 11.

The denominator is the bottom number in a fraction. It tells you what you are dividing by.

Some fractions give decimals that stop. Others give decimals that repeat forever.

Definition

Terminating and recurring decimals

A terminating decimal stops after a fixed number of decimal places, such as 0.25.
A recurring decimal has a digit or block of digits that repeats forever, such as $0.7˙0.\dot{7}$ .

Recurring notation

A dot above one digit means that digit repeats forever:

0.7˙=0.777…0.\dot{7}=0.777\ldots

Two dots show the first and last digit in the repeating block:

0.3˙6˙=0.363636…0.\dot{3}\dot{6}=0.363636\ldots

and

0.24˙9˙=0.2494949…0.2\dot{4}\dot{9}=0.2494949\ldots

Here, the 2 does not repeat; only the block 49 repeats.

Example

Convert $\frac{7}{11}$ to a decimal

Think of the fraction as a division: 7 divided by 11.
11 does not go into 7, so start with 0 point and divide 70 by 11. This gives 6 with remainder 4.
Now divide 40 by 11. This gives 3 with remainder 7.
The remainder 7 has appeared again, so the same digits will repeat. The answer is:
$711=0.636363…=0.6˙3˙\frac{7}{11}=0.636363\ldots=0.\dot{6}\dot{3}$

Tip

Spot the repeat

In long division, when the same remainder appears again, the decimal digits from that point will repeat forever.

2. The algebra trick

To turn a recurring decimal into a fraction, we usually use algebra.

The main idea is to call the decimal $x$ , multiply by a power of 10, then subtract so the repeating part cancels.

A power of 10 means numbers such as 10, 100, 1000, and so on. Multiplying by 10 moves every digit one place to the left; multiplying by 100 moves every digit two places to the left.

The picture below shows why subtracting works: once the recurring tails match, the infinite repeating part disappears.

Diagram showing recurring decimal tails lining up before subtraction

Key Idea

The cancellation trick

Choose two multiples of your decimal so that their recurring tails are identical. When you subtract, the repeating digits cancel, leaving a normal equation to solve.

3. One repeating digit straight after the decimal point

A pure recurring decimal starts repeating immediately after the decimal point.

For example, $0.6˙0.\dot{6}$ means 0.666...

Example

Prove algebraically that $0. \dot{6} = \frac{2}{3}$

Let $x$ be the recurring decimal.
$x=0.666…x=0.666\ldots$
Multiply by 10 so the repeating digits line up.
$10x=6.666…10x=6.666\ldots$
Subtract the original equation from the new equation.
$10x−x=6.666…−0.666…9x=6\begin{aligned} 10x-x&=6.666\ldots-0.666\ldots\\ 9x&=6 \end{aligned}$
Divide by 9 and simplify.
$x=69=23x=\frac{6}{9}=\frac{2}{3}$

A fraction is in simplest form when the top and bottom numbers have no common factor other than 1. A common factor is a number that divides both exactly.

4. A non-repeating digit before the repeat

Some decimals have a digit after the decimal point that does not repeat, followed by a digit that does repeat.

For example, $0.38˙=0.3888…0.3\dot{8}=0.3888\ldots$

Here, the 3 is non-recurring, and the 8 recurs.

Tip

Which powers of 10?

If there are $n$ non-recurring decimal digits and the recurring block has length $r$ , compare $10^{n+r}x$ with $10^n x$ . This lines up the recurring tails.

Example

Write $0.3 \dot{8}$ as a fraction in its simplest form

Let $x$ be the decimal.
$x=0.3888…x=0.3888\ldots$
The recurring part starts after one decimal digit, so first multiply by 10.
$10x=3.888…10x=3.888\ldots$
One full recurring digit later, multiply by 100.
$100x=38.888…100x=38.888\ldots$
Subtract the two equations so the recurring 8s cancel.
$100x−10x=38.888…−3.888…90x=35\begin{aligned} 100x-10x&=38.888\ldots-3.888\ldots\\ 90x&=35 \end{aligned}$
Solve for $x$ and simplify.
$x=3590=718x=\frac{35}{90}=\frac{7}{18}$

Common Mistake

Multiplying by the wrong amount

For a decimal like $0.38˙0.3\dot{8}$ , do not just multiply by 10 and stop. The clean exam method is to compare $100 x$ with $10 x$ , because both have the same recurring tail.

5. Repeating blocks with more than one digit

If the repeating block has 2 digits, multiply by 100. If it has 3 digits, multiply by 1000.

Example

Write $0. \dot{4} \dot{7}$ as a fraction

Let $x$ be the decimal.
$x=0.474747…x=0.474747\ldots$
The repeating block has 2 digits, so multiply by 100.
$100x=47.474747…100x=47.474747\ldots$
Subtract the original equation.
$100x−x=47.474747…−0.474747…99x=47\begin{aligned} 100x-x&=47.474747\ldots-0.474747\ldots\\ 99x&=47 \end{aligned}$
Divide by 99.
$x=4799x=\frac{47}{99}$

6. Whole numbers before the decimal point

Example

Write $3.2 \dot{6} \dot{3}$ as a fraction in its simplest form

Write the decimal out so the pattern is clear.
$3.26˙3˙=3.263636…3.2\dot{6}\dot{3}=3.263636\ldots$
Let $x$ equal the decimal.
$x=3.263636…x=3.263636\ldots$
The non-recurring digit is 2, and the recurring block is 63. So compare $1000 x$ with $10 x$ .
$1000x=3263.636363…10x=32.636363…\begin{aligned} 1000x&=3263.636363\ldots\\ 10x&=32.636363\ldots \end{aligned}$
Subtract.
$1000x−10x=3263.636363…−32.636363…990x=3231\begin{aligned} 1000x-10x&=3263.636363\ldots-32.636363\ldots\\ 990x&=3231 \end{aligned}$
Divide by 990 and simplify.
$x=3231990=359110x=\frac{3231}{990}=\frac{359}{110}$

Tip

Keep the whole number inside $x$

You do not need to split off the whole number first. Let $x$ equal the entire decimal and use the same subtraction method.

7. Using a letter for a digit

Sometimes a question uses a letter to represent a digit.

An integer is a whole number, such as 0, 1, 2, or 9. If a digit $a$ is between 1 and 9, then it can stand for any one of those whole-number digits.

Example

Prove that $0. \dot{0} \overset{a}{˙} = \frac{a}{99}$ for a digit $a$

The dots show that the block 0a repeats forever.
$x=0.0a0a0a…x=0.0a0a0a\ldots$
The repeating block has 2 digits, so multiply by 100.
$100x=a.0a0a…100x=a.0a0a\ldots$
Subtract the original decimal.
$100x−x=a.0a0a…−0.0a0a…99x=a\begin{aligned} 100x-x&=a.0a0a\ldots-0.0a0a\ldots\\ 99x&=a \end{aligned}$
Divide by 99.
$x=a99x=\frac{a}{99}$

8. Calculations with recurring decimals

For multiplication or division questions, do not try to multiply the recurring decimals directly. Convert each recurring decimal into a fraction first.

Example

Work out $0.4 \dot{2} \div 0. \dot{3}$

First convert $0.42˙0.4\dot{2}$ into a fraction.
$x=0.4222…100x=42.222…10x=4.222…90x=38x=3890=1945\begin{aligned} x&=0.4222\ldots\\ 100x&=42.222\ldots\\ 10x&=4.222\ldots\\ 90x&=38\\ x&=\frac{38}{90}=\frac{19}{45} \end{aligned}$
Convert $0.3˙0.\dot{3}$ into a fraction.
$0.3˙=130.\dot{3}=\frac{1}{3}$
Replace the recurring decimals with fractions.
$0.42˙÷0.3˙=1945÷130.4\dot{2}\div 0.\dot{3}=\frac{19}{45}\div\frac{1}{3}$
Dividing by a fraction means multiplying by its reciprocal.
$1945÷13=1945×31=1915\frac{19}{45}\div\frac{1}{3}=\frac{19}{45}\times\frac{3}{1}=\frac{19}{15}$

Exam technique

In the exam

Write the recurring decimal out with a few extra digits so you can see exactly what repeats.
Choose powers of 10 that make the recurring tails match, then subtract the equations.
Always simplify your final fraction; for calculations, convert every recurring decimal before multiplying or dividing.

Self review

Check yourself

Can you explain why $100 x$ and $10 x$ are compared for a decimal like $0.74˙0.7\dot{4}$ ?
Can you convert $0.5˙0.\dot{5}$ into a fraction using algebra?
Can you work out $0.21˙×0.3˙0.2\dot{1}\times 0.\dot{3}$ by converting both decimals first?

Recurring Decimals to Fractions

What you'll learn

1. Fractions and decimals

Recurring notation

2. The algebra trick

3. One repeating digit straight after the decimal point

4. A non-repeating digit before the repeat

5. Repeating blocks with more than one digit

6. Whole numbers before the decimal point

7. Using a letter for a digit

8. Calculations with recurring decimals

Recap questions

You've reached the end

Recurring Decimals to Fractions

What you'll learn

1. Fractions and decimals

Recurring notation

2. The algebra trick

3. One repeating digit straight after the decimal point

4. A non-repeating digit before the repeat

5. Repeating blocks with more than one digit

6. Whole numbers before the decimal point

7. Using a letter for a digit

8. Calculations with recurring decimals

Recap questions

You've reached the end

Recurring Decimals to Fractions

What you'll learn

1. Fractions and decimals

Recurring notation

2. The algebra trick

3. One repeating digit straight after the decimal point

4. A non-repeating digit before the repeat

5. Repeating blocks with more than one digit

6. Whole numbers before the decimal point

7. Using a letter for a digit

8. Calculations with recurring decimals

Recap questions

You've reached the end

Recurring Decimals to Fractions Revision Guide

Recurring Decimals to Fractions

What you'll learn

1. Fractions and decimals

Recurring notation

2. The algebra trick

3. One repeating digit straight after the decimal point

4. A non-repeating digit before the repeat

5. Repeating blocks with more than one digit

6. Whole numbers before the decimal point

7. Using a letter for a digit

8. Calculations with recurring decimals

Recap questions

You've reached the end

Recurring Decimals to Fractions Revision Guide