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Write the repeating decimal as a fraction. $\newline$ $.556556556$

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Write a repeating decimal as a fraction

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Q. Write the repeating decimal as a fraction.

\newline

.556556556

Assign Variable: Let's represent the repeating decimal $0.556556556\ldots$ with a variable. $\newline$ Let $x = 0.556556556\ldots$
Multiply by $1000$ : To isolate the repeating part, we'll multiply $x$ by $1000$ , since the repeating part has three digits. $\newline$ $1000x = 556.556556556\ldots$
Subtract Original: Now, we subtract the original $x$ from $1000x$ to get rid of the decimal part. $\newline$ $1000x - x = 556.556556556... - 0.556556556...$ $\newline$ This simplifies to: $\newline$ $999x = 556$
Divide by $999$ : To find the value of $x$ , we divide both sides of the equation by $999$ . $x = \frac{556}{999}$
Simplify Fraction: We can simplify the fraction by looking for the greatest common divisor (GCD) of $556$ and $999$ . The GCD of $556$ and $999$ is $1$ , so the fraction is already in its simplest form.
Simplify Fraction: We can simplify the fraction by looking for the greatest common divisor (GCD) of $556$ and $999$ . The GCD of $556$ and $999$ is $1$ , so the fraction is already in its simplest form.Therefore, the repeating decimal $0.556556556\ldots$ as a fraction is $\frac{556}{999}$ .

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