Recurring Decimals to Fractions

Recurring decimals can be written exactly as fractions. To do this, let the recurring decimal equal \(x\), multiply by powers of 10 to line up the recurring part, then subtract to remove the recurring digits. Solve the equation and write the final answer as a fraction in its simplest form.

Common mistake: Make sure you multiply by the correct power of \(10\) so the recurring digits line up. For example, for \(0.\dot{2}\dot{7}\), use \(100x\), not just \(10x\).

Worked example:
Write \(0.\dot{6}\) as a fraction.
  1. Let \(x = 0.666\ldots\).
    \[10x = 6.666\ldots\]
    \[10x - x = 6\]
    \[9x = 6\]
    \[x = \dfrac{6}{9} = \dfrac{2}{3}\]
Answer:
\[0.\dot{6} = \dfrac{2}{3}\]
Worked example:
Write \(0.4\dot{3}\dot{6}\) as a fraction.
  1. Let \(x = 0.4363636\ldots\).
    \[10x = 4.363636\ldots\]
    \[1000x = 436.363636\ldots\]
    \[1000x - 10x = 432\]
    \[990x = 432\]
    \[x = \dfrac{432}{990} = \dfrac{24}{55}\]
Answer:
\[0.4\dot{3}\dot{6} = \dfrac{24}{55}\]
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