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Sequences & Series

Question 5 of 8

Geometric Progression, Series & Sums

Question

Express the following recurring decimal as a fraction in its lowest terms:

\(0.\dot{0}7692\dot{3}\)

Show solution Hide solution Fully worked — 7 steps
  1. Equivalent of the recurring decimal
    \[ 0.\dot{0}7692\dot{3} = 0.076923076923\ldots \]
  2. Let \(x\) equal the decimal
    \[ \text{Let } x = 0.076923076923\ldots \qquad [1] \]
  3. Multiply \(x\) and \(0.076923076923\ldots\) by \(1000000\)
    \[ 1000000x = 76923.076923\ldots \qquad [2] \]
  4. Subtract the equation [1] from [2]
    \[ 1000000x - x = 76923.076923\ldots - 0.076923\ldots \]
  5. Evaluate both sides of the equation
    \[ 999999x = 76923 \]
  6. Divide both sides by \(999999\)
    \[ x = \frac{76923}{999999} \]
  7. Simplify \(\frac{76923}{999999}\)
    \[ x = \frac{1}{13} \]

    That is, as a fraction in its lowest terms, the recurring decimal is one thirteenth.