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Write the repeating decimal as a fraction.\newline.771771771.771771771

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Full solution

Q. Write the repeating decimal as a fraction.\newline.771771771.771771771
  1. Define xx as repeating decimal: Let xx be the repeating decimal we want to convert to a fraction.x=0.771771771…x = 0.771771771\ldots
  2. Multiply by 10001000: Multiply xx by 10001000 to shift the decimal point three places to the right, since the repeating pattern has three digits (771)(771).\newline1000x=771.771771…1000x = 771.771771\ldots
  3. Subtract original number: Subtract the original number xx from the result of the multiplication to eliminate the repeating part.1000x−x=771.771771…−0.771771771…1000x - x = 771.771771\ldots - 0.771771771\ldots
  4. Perform subtraction: Perform the subtraction on the left side of the equation. 1000x−x=999x1000x - x = 999x
  5. Combine results in equation: Perform the subtraction on the right side of the equation. 771.771771…−0.771771771…=771771.771771\ldots - 0.771771771\ldots = 771
  6. Divide by 999999: Combine the results of the subtraction to form an equation.\newline999x=771999x = 771
  7. Simplify fraction: Divide both sides of the equation by 999999 to solve for xx. \newlinex=771999x = \frac{771}{999}
  8. Divide by GCD: Simplify the fraction by finding the greatest common divisor (GCD) of 771771 and 999999. The GCD of 771771 and 999999 is 33. \newlinex=771/3999/3x = \frac{771 / 3}{999 / 3}
  9. Divide by GCD: Simplify the fraction by finding the greatest common divisor (GCD) of 771771 and 999999. The GCD of 771771 and 999999 is 33. \newlinex=(7713)/(9993)x = (\frac{771}{3}) / (\frac{999}{3}) Divide both the numerator and the denominator by the GCD to get the simplified fraction. \newlinex=257333x = \frac{257}{333}

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