To convert the repeating decimal [tex]$0.5\overline{83}$[/tex] to a fraction, we will follow a series of algebraic steps.
1. Assign the repeating decimal to a variable:
Let [tex]\( x = 0.5\overline{83} \)[/tex], which means [tex]\( x = 0.5838383\ldots \)[/tex].
2. Multiply by a power of 10 to move the decimal point to the right such that the repeating part aligns:
Multiply [tex]\( x \)[/tex] by 1000 (since the repeating part has two digits):
[tex]\[
1000x = 583.838383\ldots
\][/tex]
3. Set up the original number as another equation:
We already have:
[tex]\[
x = 0.5838383\ldots
\][/tex]
4. Subtract the original equation from this new equation to eliminate the repeating part:
[tex]\[
1000x = 583.838383\ldots \\
x = 0.5838383\ldots \\
\therefore 999x = 583.838383\ldots - 0.5838383\ldots \\
999x = 583.25
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
999x = 583.25 \\
x = \frac{583.25}{999}
\][/tex]
6. Simplify the fraction:
To simplify [tex]\(\frac{583.25}{999}\)[/tex], we can first convert the decimal to a fraction:
[tex]\[
583.25 = 583 \frac{1}{4} = 583 \frac{25}{100} = 583 \frac{1}{4}
\][/tex]
So, our fraction becomes:
[tex]\[
\frac{583 \frac{1}{4}}{999} = \frac{2333}{3996}
\][/tex]
7. Simplify the final fraction:
Now, simplify [tex]\(\frac{2333}{3996}\)[/tex]. Note that the greatest common divisor (GCD) of 2333 and 3996 might not be obvious, so we use the Euclidean algorithm or a factorization approach, but noticing the numbers are larger, we can verify:
[tex]\[ \frac{2333}{3996} \text{ can be simplified to:} \frac{7}{12} \][/tex]
Therefore, the fraction representation for the repeating decimal [tex]\(0.5\overline{83}\)[/tex] is [tex]\[
\boxed{\frac{7}{12}}
\][/tex]
