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Write the repeating decimal as a fraction.\newline.683683683.683683683

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Full solution

Q. Write the repeating decimal as a fraction.\newline.683683683.683683683
  1. Identify Repeating Pattern: Let's identify the repeating pattern in the decimal. The digits “683”\text{“683”} repeat indefinitely.
  2. Convert to Fraction: To convert the repeating decimal to a fraction, let's denote the repeating decimal as xx:x=0.683683683…x = 0.683683683\ldots
  3. Isolate Repeating Part: To isolate the repeating part, we can multiply xx by 10001000, since the repeating part has three digits: 1000x=683.683683683…1000x = 683.683683683\ldots
  4. Subtract Decimal: Now, we subtract the original xx from 1000x1000x to get rid of the decimal part:\newline1000x−x=683.683683683...−0.683683683...1000x - x = 683.683683683... - 0.683683683...\newline999x=683999x = 683
  5. Divide by 999999: To find the value of xx, we divide both sides of the equation by 999999:x=683999x = \frac{683}{999}
  6. Simplify Fraction: We can simplify the fraction by looking for common factors of the numerator and the denominator. Both 683683 and 999999 are divisible by 11, and there are no other common factors since 683683 is a prime number.
  7. Final Fraction: Therefore, the fraction is already in its simplest form: x=683999x = \frac{683}{999}

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