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Write the repeating decimal as a fraction.\newline.74747474.74747474

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Full solution

Q. Write the repeating decimal as a fraction.\newline.74747474.74747474
  1. Denote Repeating Decimal: Let's denote the repeating decimal 0.74747474…0.74747474\ldots as xx.x=0.74747474…x = 0.74747474\ldots
  2. Convert to Fraction: To convert this repeating decimal into a fraction, we first express it as an infinite sum of its repeating parts.\newline0.74747474…=0.74+0.0074+0.000074+…0.74747474\ldots = 0.74 + 0.0074 + 0.000074 + \ldots
  3. Express as Infinite Sum: Notice that each term after the first is 1100\frac{1}{100} times the previous term, which means this is a geometric series with the first term a=0.74a = 0.74 and the common ratio r=1100.r = \frac{1}{100}.
  4. Identify Geometric Series: The sum SS of an infinite geometric series with first term aa and common ratio rr (where ∣r∣<1|r| < 1) is given by S=a1−rS = \frac{a}{1 - r}.
  5. Apply Sum Formula: We apply the formula to find the sum of the series, which represents our repeating decimal. S=0.741−1100S = \frac{0.74}{1 - \frac{1}{100}}
  6. Perform Calculation: Now we perform the calculation.\newlineS=0.7499/100S = \frac{0.74}{99/100}\newlineS=0.74×10099S = 0.74 \times \frac{100}{99}
  7. Write Decimal as Fraction: To avoid dealing with decimals, we can write 0.740.74 as 74100\frac{74}{100}.S=(74100)×(10099)S = \left(\frac{74}{100}\right) \times \left(\frac{100}{99}\right)
  8. Simplify Fraction: We simplify the fraction by canceling out the common factor of 100100.S=7499S = \frac{74}{99}
  9. Final Answer: The fraction 7499\frac{74}{99} cannot be simplified further, so this is our final answer.

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