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Write the repeating decimal as a fraction.\newline0.6676676670.667667667

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Full solution

Q. Write the repeating decimal as a fraction.\newline0.6676676670.667667667
  1. Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The digits ＂667667＂ repeat indefinitely.\newlinePattern identified: 0.667667667…=0.667+0.000667+0.000000667+…0.667667667\ldots = 0.667 + 0.000667 + 0.000000667 + \ldots
  2. Express Terms as Fractions: Express each term in the pattern as a fraction.\newline0.667667667...=0.667+0.000667+0.000000667+...0.667667667... = 0.667 + 0.000667 + 0.000000667 + ...\newline=6671000+6671000000+6671000000000+...= \frac{667}{1000} + \frac{667}{1000000} + \frac{667}{1000000000} + ...
  3. Recognize Geometric Series: Recognize that the series 6671000+6671000000+6671000000000+…\frac{667}{1000} + \frac{667}{1000000} + \frac{667}{1000000000} + \ldots forms a geometric series.\newlineFind the common ratio (r)(r) of the geometric series by dividing two consecutive terms.\newline(6671000000)/(6671000)=6671000000×1000667=11000(\frac{667}{1000000}) / (\frac{667}{1000}) = \frac{667}{1000000} \times \frac{1000}{667} = \frac{1}{1000}\newlineCommon Ratio (r):11000(r): \frac{1}{1000}
  4. Find Common Ratio: Write the repeating decimal as a fraction using the formula for the sum of an infinite geometric series, which is a1/(1−r)a_1 / (1 - r), where a1a_1 is the first term.\newlineSubstitute a1=667/1000a_1 = 667/1000 and r=1/1000r = 1/1000 into the formula.\newline(667/1000)/(1−1/1000)=(667/1000)/(999/1000)=667/999(667/1000) / (1 - 1/1000) = (667/1000) / (999/1000) = 667/999
  5. Write as Fraction: Simplify the fraction.\newline6671000×1000999=667999\frac{667}{1000} \times \frac{1000}{999} = \frac{667}{999}\newlineSo, 0.667667667…=6679990.667667667\ldots = \frac{667}{999}

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