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Write the repeating decimal as a fraction. $\newline$ $.855855855$

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Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.855855855

Identify Repeating Sequence: Identify the repeating sequence in the decimal. $\newline$ The repeating sequence in the decimal $0.855855855$ is $855$ . This sequence repeats indefinitely, so we can denote the repeating decimal as $0.\overline{855}$ .
Assign Variable: Let $x$ equal the repeating decimal. $\newline$ Let $x = 0.\overline{855}$ .
Multiply by Power of $10$ : Multiply $x$ by a power of $10$ that will move the decimal point to the right so that the repeating sequence aligns with the original decimal. $\newline$ Since the repeating sequence is three digits long, we multiply $x$ by $10^3$ (which is $1000$ ) to move the decimal point three places to the right. $\newline$ $1000x = 855.\overline{855}$ .
Set Up Equation: Set up an equation to eliminate the repeating part. $\newline$ We now have two expressions: $x = 0.\overline{855}$ and $1000x = 855.\overline{855}$ . By subtracting the first equation from the second, we can eliminate the repeating part. $\newline$ $1000x - x = 855.\overline{855} - 0.\overline{855}$ .
Perform Subtraction: Perform the subtraction to solve for $x$ . $1000x - x = 855.\overline{855} - 0.\overline{855}$ simplifies to $999x = 855$ .
Divide by $999$ : Divide both sides of the equation by $999$ to solve for $x$ . $x = \frac{855}{999}$ .
Simplify Fraction: Simplify the fraction. $\newline$ To simplify the fraction $\frac{855}{999}$ , we need to find the greatest common divisor (GCD) of $855$ and $999$ . The GCD of $855$ and $999$ is $9$ .
Simplify Fraction: Simplify the fraction. $\newline$ To simplify the fraction $\frac{855}{999}$ , we need to find the greatest common divisor (GCD) of $855$ and $999$ . The GCD of $855$ and $999$ is $9$ .Divide both the numerator and the denominator by the GCD to get the simplified fraction. $\newline$ $855 \div 9 = 95$ and $999 \div 9 = 111$ , so $x = \frac{95}{111}$ .

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