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$Convert the following repeating decimal to a fraction in simplest form. . bar(15) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $. \overline{15}$ $\newline$ Answer:

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Q. Convert the following repeating decimal to a fraction in simplest form.

\newline

. \overline{15}

\newline

Answer:

Set up equation with $x$ : Let $x = 0.151515...$ $\newline$ To convert the repeating decimal to a fraction, we can set up an equation using $x$ to represent the repeating decimal. Since the digits $15$ repeat, we will multiply $x$ by $100$ to shift the decimal two places to the right.
Multiply by $100$ : Multiplying $x$ by $100$ , we get: $\newline$ $100x = 15.151515...$ $\newline$ Now we have two expressions: $x = 0.151515...$ and $100x = 15.151515...$ $\newline$ Subtracting the first equation from the second will help us eliminate the repeating part.
Subtract equations: Subtract the first equation from the second: $\newline$ $100x - x = 15.151515... - 0.151515...$ $\newline$ $99x = 15$ $\newline$ Now we can solve for $x$ by dividing both sides of the equation by $99$ .
Solve for x: Dividing both sides by $99$ , we get: $\newline$ $x = \frac{15}{99}$ $\newline$ Now we need to simplify the fraction to its simplest form.
Simplify the fraction: To simplify the fraction, we find the greatest common divisor (GCD) of $15$ and $99$ . The GCD of $15$ and $99$ is $3$ . Dividing both the numerator and the denominator by $3$ , we get: $x = (\frac{15}{3}) / (\frac{99}{3})$ $x = \frac{5}{33}$