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Write the repeating decimal as a fraction.\newline.168168168.168168168

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Full solution

Q. Write the repeating decimal as a fraction.\newline.168168168.168168168
  1. Assign xx value: Let xx equal the repeating decimal 0.168168168…0.168168168\ldots\newlinex=0.168168168…x = 0.168168168\ldots
  2. Multiply by 10001000: Multiply xx by 10001000 to shift the decimal point three places to the right, since the repeating block has three digits (168168).\newline1000x=168.168168168…1000x = 168.168168168\ldots
  3. Subtract equations: Subtract the original equation x=0.168168168...x = 0.168168168... from the new equation 1000x=168.168168168...1000x = 168.168168168... to eliminate the repeating decimals.\newline1000x−x=168.168168168...−0.168168168...1000x - x = 168.168168168... - 0.168168168...
  4. Find value of 999x999x: Perform the subtraction to find the value of 999x999x.999x=168999x = 168
  5. Divide by 999999: Divide both sides of the equation by 999999 to solve for xx.x=168999x = \frac{168}{999}
  6. Simplify fraction: Simplify the fraction by finding the greatest common divisor (GCD) of 168168 and 999999. The GCD of 168168 and 999999 is 33. \newlinex=168/3999/3x = \frac{168 / 3}{999 / 3}
  7. Simplify fraction: Simplify the fraction by finding the greatest common divisor (GCD) of 168168 and 999999. The GCD of 168168 and 999999 is 33.
    x=1683/9993x = \frac{168}{3} / \frac{999}{3} Divide both the numerator and the denominator by the GCD to get the simplified fraction.
    x=56333x = \frac{56}{333}

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