Sure! Let's express the repeating decimal [tex]\( 0 . \overline{83} \)[/tex] as a fraction in simplest form.
1. Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[
x = 0.838383 \ldots
\][/tex]
2. To eliminate the repeating part, multiply [tex]\( x \)[/tex] by 100:
[tex]\[
100x = 83.838383 \ldots
\][/tex]
3. Now you have two equations:
[tex]\[
x = 0.838383 \ldots
\][/tex]
[tex]\[
100x = 83.838383 \ldots
\][/tex]
4. Subtract the first equation from the second equation to eliminate the repeating decimal:
[tex]\[
100x - x = 83.838383 \ldots - 0.838383 \ldots
\][/tex]
[tex]\[
99x = 83
\][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides by 99:
[tex]\[
x = \frac{83}{99}
\][/tex]
6. Simplify the fraction if possible. In this case, [tex]\( \frac{83}{99} \)[/tex] is already in simplest form because the greatest common divisor (GCD) of 83 and 99 is 1.
Therefore, the repeating decimal [tex]\( 0 . \overline{83} \)[/tex] expressed as a fraction in simplest form is:
[tex]\[
\boxed{\frac{83}{99}}
\][/tex]
