Fractions and Decimals • Topic 5 of 7

Recurring Decimals

A recurring (repeating) decimal has a block of digits that repeats forever, written with a bar or dots, such as 0.333… = 1/3 or 0.1666… = 1/6. A fraction repeats exactly when its denominator in lowest terms has a prime factor other than 2 or 5. To turn a simple repeating decimal into a fraction, use the shortcut: a single repeating digit d gives d/9, a two-digit block gives the block over 99, and so on (so 0.͞27͞… = 27/99 = 3/11). Recognising 1/3, 2/3 and 1/9 instantly is a useful SAT time-saver.

✅ Solved examples

1. Write 0.333… as a fraction.
A single repeating digit 3 over 9 gives 3/9 = 1/3.
2. Write 0.666… as a fraction.
6/9 = 2/3.
3. Does 1/6 give a recurring decimal?
6 = 2·3 has the factor 3, so yes: 1/6 = 0.1666….
4. Write the repeating decimal 0.272727… as a fraction.
A two-digit repeating block 27 over 99 gives 27/99 = 3/11.

✏️ Practice — try these, take hints as needed

1. Write 0.555… as a fraction.
One repeating digit goes over 9.
Write 5/9.
Check if it reduces (it does not).
5/9.
2. Write 0.999… as a fraction.
One repeating digit 9 over 9.
9/9.
Simplify.
1.
3. Does 5/12 give a terminating or recurring decimal?
Factor 12 = 2² · 3.
The factor 3 is not 2 or 5.
Such fractions repeat.
Recurring (0.41666…).
4. Write 0.121212… as a fraction.
Two-digit block 12 over 99.
12/99.
Divide both by 3.
4/33.
5. Write 0.444… as a fraction.
Single digit 4 over 9.
4/9.
Already in lowest terms.
4/9.

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Converting Fractions to Decimals Comparing Fractions →