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Write the repeating decimal as a fraction.\newline.993993993.993993993

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Full solution

Q. Write the repeating decimal as a fraction.\newline.993993993.993993993
  1. Identify Repeating Pattern: Let's identify the repeating pattern in the decimal 0.993993993…0.993993993\ldots The repeating pattern is 993993.
  2. Express as Sum: Let's express the repeating decimal as a sum of its repeating parts.\newline.993993993...=0.993+0.000993+0.000000993+....993993993... = 0.993 + 0.000993 + 0.000000993 + ...
  3. Express as Fraction: Now, let's express each term as a fraction.\newline0.993=99310000.993 = \frac{993}{1000}\newline0.000993=99310000000.000993 = \frac{993}{1000000}\newline0.000000993=99310000000000.000000993 = \frac{993}{1000000000}\newline...\newlineThis is a geometric series with the first term a1=9931000a_1 = \frac{993}{1000} and the common ratio r=11000r = \frac{1}{1000}.
  4. Use Geometric Series Formula: To find the sum of an infinite geometric series, we use the formula S=a11−rS = \frac{a_1}{1 - r}, where SS is the sum, a1a_1 is the first term, and rr is the common ratio.\newlineSubstitute a1=9931000a_1 = \frac{993}{1000} and r=11000r = \frac{1}{1000} into the formula.\newlineS=99310001−11000S = \frac{\frac{993}{1000}}{1 - \frac{1}{1000}}
  5. Perform Calculation: Now, let's perform the calculation.\newlineS=9931000/9991000S = \frac{993}{1000} / \frac{999}{1000}\newlineS=9931000×1000999S = \frac{993}{1000} \times \frac{1000}{999}\newlineS=993999S = \frac{993}{999}
  6. Simplify Fraction: We can simplify the fraction 993999\frac{993}{999} by dividing both the numerator and the denominator by their greatest common divisor, which is 33. \newlineS=(993÷3999÷3)S = (\frac{993 \div 3}{999 \div 3})\newlineS=331333S = \frac{331}{333}

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