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Write the repeating decimal as a fraction.\newline.47474747.47474747

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Full solution

Q. Write the repeating decimal as a fraction.\newline.47474747.47474747
  1. Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The digits ＂4747＂ repeat indefinitely.\newlinePattern identified: 0.47474747…=0.47+0.0047+0.000047+…0.47474747\ldots = 0.47 + 0.0047 + 0.000047 + \ldots
  2. Express Terms as Fractions: Express each term in the pattern as a fraction.\newline0.47474747…=47100+4710000+471000000+…0.47474747\ldots = \frac{47}{100} + \frac{47}{10000} + \frac{47}{1000000} + \ldots
  3. Recognize Geometric Series: Recognize that the series 47100+4710000+471000000+…\frac{47}{100} + \frac{47}{10000} + \frac{47}{1000000} + \ldots is a geometric series.\newlineTo find the common ratio (r)(r), we divide a term by the term before it.\newline(4710000)/(47100)=4710000×10047=1100(\frac{47}{10000}) / (\frac{47}{100}) = \frac{47}{10000} \times \frac{100}{47} = \frac{1}{100}\newlineCommon Ratio (r):1100(r): \frac{1}{100}
  4. Write as Fraction Using Formula: 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=47/100a_1 = 47/100 and r=1/100r = 1/100 into the formula.\newline(47/100)/(1−1/100)=(47/100)/(99/100)=47/100×100/99(47/100) / (1 - 1/100) = (47/100) / (99/100) = 47/100 \times 100/99
  5. Simplify the Fraction: Simplify the fraction. 47100×10099=4799\frac{47}{100} \times \frac{100}{99} = \frac{47}{99} So, 0.47474747…=47990.47474747\ldots = \frac{47}{99}

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