To convert the repeating decimal [tex]\(0.8\overline{3}\)[/tex] into a simplified fraction, we'll follow these steps:
1. Let [tex]\(x\)[/tex] be the repeating decimal:
[tex]\[
x = 0.8\overline{3} = 0.83333\dots
\][/tex]
2. Multiply [tex]\(x\)[/tex] by 10 to shift the decimal point:
[tex]\[
10x = 8.33333\dots
\][/tex]
3. Set up the equations:
We now have two equations:
[tex]\[
x = 0.83333\dots \quad \text{(Equation 1)}
\][/tex]
[tex]\[
10x = 8.33333\dots \quad \text{(Equation 2)}
\][/tex]
4. Subtract Equation 1 from Equation 2 to eliminate the repeating part:
[tex]\[
10x - x = 8.33333\dots - 0.83333\dots
\][/tex]
This simplifies to:
[tex]\[
9x = 7.5
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{7.5}{9}
\][/tex]
6. Simplify the fraction by finding the greatest common divisor (GCD) for the numerator and the denominator:
[tex]\[
\frac{7.5}{9} = \frac{75}{90} \quad \text{(multiplying by 10 to clear the decimal)}
\][/tex]
The GCD of 75 and 90 is 15. Now we can simplify the fraction by dividing both the numerator and the denominator by their GCD:
[tex]\[
\frac{75 \div 15}{90 \div 15} = \frac{5}{6}
\][/tex]
7. Conclusion:
The simplified fraction that represents the repeating decimal [tex]\(0.8\overline{3}\)[/tex] is:
[tex]\[
0.8\overline{3} = \frac{5}{6}
\][/tex]
