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Write the repeating decimal as a fraction.\newline.833833833.833833833

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Full solution

Q. Write the repeating decimal as a fraction.\newline.833833833.833833833
  1. Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The repeating pattern is 833833.
  2. Express Decimal as Sum: Express the repeating decimal as a sum of its parts: 0.833833833…=0.833+0.000833+0.000000833+…0.833833833\ldots = 0.833 + 0.000833 + 0.000000833 + \ldots
  3. Convert to Fractions: Convert each part into a fraction: 0.833=83310000.833 = \frac{833}{1000}, and each subsequent part is 10001000 times smaller than the previous one.
  4. Recognize Geometric Series: Recognize that this is a geometric series with the first term a1=8331000a_1 = \frac{833}{1000} and the common ratio r=11000r = \frac{1}{1000}.
  5. Use Infinite Series Formula: Use the formula for the sum of an infinite geometric series, S=a11−rS = \frac{a_1}{1 - r}, to find the fraction equivalent of the repeating decimal.
  6. Substitute Values: Substitute the values of a1a_1 and rr into the formula: S=8331000/(1−11000)S = \frac{833}{1000} / \left(1 - \frac{1}{1000}\right).
  7. Simplify Denominator: Simplify the denominator: 1−11000=99910001 - \frac{1}{1000} = \frac{999}{1000}.
  8. Calculate the Sum: Now, calculate the sum: S=8331000/9991000=8331000×1000999S = \frac{833}{1000} / \frac{999}{1000} = \frac{833}{1000} \times \frac{1000}{999}.
  9. Simplify the Fraction: Simplify the fraction: S=833999S = \frac{833}{999}.

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