Recurring Decimals

Method — Align, subtract, simplify

1. Let x equal the exact recurring decimal.

2. Multiply by powers of ten to create matching recurring tails. A non-recurring prefix may require two new equations.

3. Subtract one complete equation from the other, on both sides.

4. Divide both sides to find x.

5. Simplify the fraction and check it by division.

Worked Examples

Example 1 — Read the repeating block

A terminating decimal ends. For example, 0.25 stops after the hundredths place.

A recurring decimal has a digit or block of digits repeating forever. The dots … mean the stated pattern continues.

A bar over digits marks exactly the part that repeats.

Notation	Expanded decimal	What repeats?
0.3	0.333…	3
0.27	0.272727…	27
0.16	0.1666…	Only 6; the first 1 does not repeat
0.12	0.121212…	12

Read the bar before choosing a method. 0.16 is different from 0.16, which means 0.161616….

We will use bars for short notation and show the expanded pattern alongside them. An unmarked 0.33 terminates; it is not exactly 0.333….

Example 2 — Convert one repeating digit

Write 0.3 as an exact fraction.

Let x name the exact recurring value. It is a label for this number, not an extra quantity.

x = 0.333…

Multiply both sides by 10. One digit repeats, so this gives the same decimal tail:

10x = 3.333…

Here 10x means 10 × x. Subtract the original equation from this new equation:

10x − x = 3.333… − 0.333…

9x = 3

The decimal tails match exactly, so their difference is zero.

Divide both sides by 9:

x = $\frac{3}{9}$

Divide numerator and denominator by 3:

x = $\frac{1}{3}$

This is exact. We have not rounded the decimal.

Example 3 — Try another single digit

Convert 0.7 to a fraction.

x = 0.777…

10x = 7.777…

Subtract the first equation from the second:

10x − x = 7

9x = 7

Divide both sides by 9: x = $\frac{7}{9}$.

7 and 9 have no common factor greater than 1, so this is simplest form.

Check by division: 70 ÷ 9 gives digit 7 and remainder 7. The same remainder returns at each place, so 7 ÷ 9 = 0.777….

Example 4 — Match a two-digit block

Convert 0.27 to a fraction.

x = 0.272727…

Two digits repeat. Multiply by 100 to bring one whole copy of 27 before the decimal point:

100x = 27.272727…

Now the decimal tails are identical. Subtract:

100x − x = 27

99x = 27

Divide both sides by 99: x = $\frac{27}{99}$.

Divide the numerator and denominator by 9:

x = $\frac{3}{11}$

Check: 3 ÷ 11 begins 0.272727…, with the same two-digit block.

The matching powers of ten are 10 for one repeating digit, 100 for two and 1000 for three. The aim is always to align the tails.

Example 5 — Allow a non-recurring prefix

Convert 0.16. The first digit 1 occurs only once; only 6 repeats.

x = 0.1666…

Multiply by 10 to put the non-recurring digit before the point:

10x = 1.666…

Multiply the original equation by 100 to bring one more recurring digit before the point:

100x = 16.666…

These two decimal tails match. Subtract 10x from 100x, not x:

100x − 10x = 16.666… − 1.666…

90x = 15

Divide both sides by 90: x = $\frac{15}{90}$.

Divide top and bottom by 15:

x = $\frac{1}{6}$

Check: 1 ÷ 6 gives first digit 1 with remainder 4. Then 40 ÷ 6 gives digit 6 with remainder 4 again, so the 6 repeats.

Example 6 — Use the prefix method again

Convert 0.23 to a fraction.

x = 0.2333…

10x = 2.333…

100x = 23.333…

Subtract the equations with matching decimal tails:

100x − 10x = 23 − 2

90x = 21

x = $\frac{21}{90}$

Divide both parts by 3: x = $\frac{7}{30}$.

Do not use $\frac{23}{99}$: that would give 0.232323…, with both digits repeating.

Example 7 — Go from a fraction to a recurring decimal

A fraction means division: divide its numerator, the top number, by its denominator, the bottom number.

First convert $\frac{2}{3}$.

2 is smaller than 3, so the whole-number part is 0. At the tenths step, divide 20 by 3.

20 ÷ 3 gives digit 6 with remainder 2.

Bringing down another zero gives 20 again. The same calculation repeats.

$\frac{2}{3}$ = 0.6 = 0.666…

Now convert $\frac{1}{11}$.

Decimal place	Division step	Digit	Remainder
Tenths	10 ÷ 11	0	10
Hundredths	100 ÷ 11	9	1
Thousandths	10 ÷ 11	0	10
Ten-thousandths	100 ÷ 11	9	1

Each step brings down a zero after the previous remainder. The cycle of remainders repeats.

$\frac{1}{11}$ = 0.09 = 0.090909…

The zero belongs in the recurring block. Removing it would change the value.

Example 8 — GCSE Higher: a prefix and a two-digit block

Write 0.127 as a fraction in its simplest form. Only 27 repeats after the first 1.

x = 0.1272727…

First multiply by 10 to move the one non-recurring digit before the point:

10x = 1.272727…

There are two repeating digits, so multiply that equation by 100:

1000x = 127.272727…

Subtract the two equations with identical tails:

1000x − 10x = 127 − 1

990x = 126

x = $\frac{126}{990}$

Divide numerator and denominator by 18:

x = $\frac{7}{55}$

Check by division: 7 ÷ 55 = 0.1272727…. The first 1 is followed by the repeating block 27.