Write the repeating decimal as a fraction. .771771771 ______

Define x as repeating decimal: Let x be the repeating decimal we want to convert to a fraction. x = 0.771771771...Multiply by 1000: Multiply x by 1000 to shift the decimal point three places to the right, since the repeating pattern has three digits (771). 1000x = 771.771771...Subtract original number: Subtract the original number x from the result of the multiplication to eliminate the repeating part. 1000x - x = 771.771771... - 0.771771771...Perform subtraction: Perform the subtraction on the left side of the equation. 1000x - x = 999xCombine results in equation: Perform the subtraction on the right side of the equation. 771.771771... - 0.771771771... = 771Divide by 999: Combine the results of the subtraction to form an equation. 999x = 771Simplify fraction: Divide both sides of the equation by 999 to solve for x. x = 771 / 999Divide by GCD: Simplify the fraction by finding the greatest common divisor (GCD) of 771 and 999. The GCD of 771 and 999 is 3. x = (771 / 3) / (999 / 3)Divide both the numerator and the denominator by the GCD to get the simplified fraction. x = 257 / 333

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Algebra 2

Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

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.771771771

Define $x$ as repeating decimal: Let $x$ be the repeating decimal we want to convert to a fraction. $x = 0.771771771\ldots$
Multiply by $1000$ : Multiply $x$ by $1000$ to shift the decimal point three places to the right, since the repeating pattern has three digits $(771)$ . $\newline$ $1000x = 771.771771\ldots$
Subtract original number: Subtract the original number $x$ from the result of the multiplication to eliminate the repeating part. $1000x - x = 771.771771\ldots - 0.771771771\ldots$
Perform subtraction: Perform the subtraction on the left side of the equation. $1000x - x = 999x$
Combine results in equation: Perform the subtraction on the right side of the equation. $771.771771\ldots - 0.771771771\ldots = 771$
Divide by $999$ : Combine the results of the subtraction to form an equation. $\newline$ $999x = 771$
Simplify fraction: Divide both sides of the equation by $999$ to solve for $x$ . $\newline$ $x = \frac{771}{999}$
Divide by GCD: Simplify the fraction by finding the greatest common divisor (GCD) of $771$ and $999$ . The GCD of $771$ and $999$ is $3$ . $\newline$ $x = \frac{771 / 3}{999 / 3}$
Divide by GCD: Simplify the fraction by finding the greatest common divisor (GCD) of $771$ and $999$ . The GCD of $771$ and $999$ is $3$ . $\newline$ $x = (\frac{771}{3}) / (\frac{999}{3})$ Divide both the numerator and the denominator by the GCD to get the simplified fraction. $\newline$ $x = \frac{257}{333}$