Recurring decimals

Some fractions give decimals that repeat for ever, like 1/3 = 0.333… A dot over the digits shows which repeat. Algebra turns any recurring decimal back into a fraction.

Key facts

  • A dot over a digit (or the first and last digits of a block) means it repeats.
  • Fractions whose denominator has only 2s and 5s as prime factors terminate; others recur.
  • To convert: let x be the decimal, multiply by 10, 100… so the repeating parts line up, then subtract.

Worked examples

Example 1

Write 0.4̇ (0.444…) as a fraction.

  1. Let x = 0.444…
  2. Multiply by 10. 10x = 4.444…
  3. Subtract: the repeating parts cancel. 9x = 4
  4. So x = 4/9

Answer: 4/9

Example 2

Write 0.2̇7̇ (0.2727…) as a fraction in its simplest form.

  1. Two digits repeat, so multiply by 100. 100x = 27.2727…
  2. Subtract x. 99x = 27
  3. Simplify. x = 27/99 = 3/11

Answer: 3/11