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Write the repeating decimal as a fraction.\newline.63636363.63636363

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Full solution

Q. Write the repeating decimal as a fraction.\newline.63636363.63636363
  1. Rephrase the Problem: Let's first rephrase the ＂How can we express the repeating decimal 0.63636363…0.63636363\ldots as a fraction?＂
  2. Identify Repeating Pattern: Identify the repeating pattern in the decimal. The digits ＂6363＂ repeat indefinitely, so we can write the decimal as 0.63636363…=0.63+0.0063+0.000063+…0.63636363\ldots = 0.63 + 0.0063 + 0.000063 + \ldots
  3. Express as Fractions: Express each term in the pattern as a fraction. The decimal can be written as 0.63636363…=63100+6310000+631000000+…0.63636363\ldots = \frac{63}{100} + \frac{63}{10000} + \frac{63}{1000000} + \ldots
  4. Recognize Geometric Series: Recognize that the series 63100+6310000+631000000+…\frac{63}{100} + \frac{63}{10000} + \frac{63}{1000000} + \ldots is a geometric series with the first term a1=63100a_1 = \frac{63}{100} and a common ratio r=1100r = \frac{1}{100}.
  5. Calculate Common Ratio: Calculate the common ratio rr by dividing a term in the series by the previous term. For example, 6310000/63100=6310000×10063=1100\frac{63}{10000} / \frac{63}{100} = \frac{63}{10000} \times \frac{100}{63} = \frac{1}{100}. So, the common ratio rr is indeed 1100\frac{1}{100}.
  6. Use Infinite Series Formula: Use 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 and rr is the common ratio. Substitute a1=63100a_1 = \frac{63}{100} and r=1100r = \frac{1}{100} into the formula.
  7. Perform Calculation: Perform the calculation: (63/100)/(1−1/100)=(63/100)/(99/100)=63/100×100/99=63/99(63/100) / (1 - 1/100) = (63/100) / (99/100) = 63/100 \times 100/99 = 63/99.
  8. Simplify Fraction: Simplify the fraction 6399\frac{63}{99} by dividing both the numerator and the denominator by their greatest common divisor, which is 99. 63÷9=763 \div 9 = 7 and 99÷9=1199 \div 9 = 11.
  9. Final Fraction: The simplified fraction is 711\frac{7}{11}. Therefore, the repeating decimal 0.63636363…0.63636363\ldots can be expressed as the fraction 711\frac{7}{11}.

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