To convert \( 0.8\overline{3} \) to a fraction, follow these detailed steps:
1. Identify the repeating part and write the number as a sum of its non-repeating part and the repeating part:
[tex]\[
0.8\overline{3} = 0.83333...
\][/tex]
2. Let \( x \) represent the repeating decimal:
[tex]\[
x = 0.83333...
\][/tex]
3. Separate the non-repeating part (0.8) from the repeating part (\(\overline{3}\)) by expressing \( x \) appropriately:
[tex]\[
x = 0.8 + 0.03333...
\][/tex]
4. Let \( y \) represent the repeating part alone:
[tex]\[
y = 0.03333...
\][/tex]
5. Express \( y \) as a fraction:
[tex]\[
y = 0.03333... = \frac{1}{30} \quad \text{(since [tex]$3$[/tex] repeats every digit)}
\][/tex]
6. Combine the non-repeating part and the fraction obtained from the repeating part:
[tex]\[
x = 0.8 + \frac{1}{30}
\][/tex]
7. Express \(0.8\) as a fraction:
[tex]\[
0.8 = \frac{8}{10} = \frac{4}{5}
\][/tex]
8. Add the two fractions together, finding a common denominator if necessary:
[tex]\[
x = \frac{4}{5} + \frac{1}{30}
\][/tex]
First, find the common denominator. The least common multiple of 5 and 30 is 30.
[tex]\[
\frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}
\][/tex]
Now add:
[tex]\[
\frac{24}{30} + \frac{1}{30} = \frac{24 + 1}{30} = \frac{25}{30}
\][/tex]
9. Simplify the fraction:
[tex]\[
\frac{25}{30} = \frac{25 \div 5}{30 \div 5} = \frac{5}{6}
\][/tex]
Therefore, \( 0.8\overline{3} \) as a fraction is:
[tex]\[
\boxed{\frac{5}{6}}
\][/tex]
