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Write the repeating decimal as a fraction.\newline.313313313.313313313

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Full solution

Q. Write the repeating decimal as a fraction.\newline.313313313.313313313
  1. Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The digits “313”\text{“313”} repeat indefinitely.
  2. Convert to Fraction: To convert the repeating decimal to a fraction, let's denote the repeating decimal as xx:x=0.313313313...x = 0.313313313...
  3. Isolate Repeating Pattern: To isolate the repeating pattern, we can multiply xx by 10001000, since the pattern repeats every three decimal places: 1000x=313.313313313…1000x = 313.313313313\ldots
  4. Subtract and Simplify: Now, we subtract the original xx from 1000x1000x to get rid of the repeating decimals: 1000x−x=313.313313313...−0.313313313...1000x - x = 313.313313313... - 0.313313313... 999x=313999x = 313
  5. Solve for x: Next, we solve for x by dividing both sides of the equation by 999999:x=313999x = \frac{313}{999}
  6. Simplify Fraction: We can simplify the fraction by looking for a common divisor. In this case, the numerator and denominator do not have any common factors other than 11, so the fraction is already in its simplest form.

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