To convert the repeating decimal [tex]\(0.\overline{393}\)[/tex] into a fraction, follow these steps precisely:
1. Assign the repeating decimal to a variable:
[tex]\[
x = 0.\overline{393}
\][/tex]
2. Since there are 3 repeating digits (393), multiply both sides of the equation by [tex]\(10^3 = 1000\)[/tex]:
[tex]\[
1000x = 393.\overline{393}
\][/tex]
3. Now you have two equations:
[tex]\[
\begin{aligned}
1000x &= 393.\overline{393} \\
x &= 0.\overline{393}
\end{aligned}
\][/tex]
4. Subtract the second equation from the first equation to eliminate the repeating part:
[tex]\[
\begin{aligned}
1000x - x &= 393.\overline{393} - 0.\overline{393} \\
999x &= 393
\end{aligned}
\][/tex]
5. To solve for [tex]\(x\)[/tex], divide both sides of the equation by 999:
[tex]\[
x = \frac{393}{999}
\][/tex]
6. Simplify the fraction [tex]\(\frac{393}{999}\)[/tex] by finding the greatest common divisor (GCD) of 393 and 999.
Both 393 and 999 are divisible by 3:
[tex]\[
\frac{393 \div 3}{999 \div 3} = \frac{131}{333}
\][/tex]
The GCD of 131 and 333 is 1.
Thus, the fraction [tex]\(\frac{131}{333}\)[/tex] is already in its simplest form.
So, the value of [tex]\(x = 0.\overline{393}\)[/tex] as a fraction is:
[tex]\[
\boxed{\frac{131}{333}}
\][/tex]
