To rewrite the repeating decimal [tex]\(2.\overline{6}\)[/tex] as a simplified fraction, let's go through the steps:
1. First, identify the repeating part of the decimal. Here, it is [tex]\(0.\overline{6}\)[/tex].
2. Let [tex]\(x = 0.\overline{6}\)[/tex].
3. Since the repeating decimal part has one digit (6), multiply [tex]\(x\)[/tex] by 10 to shift the decimal point:
[tex]\[
10x = 6.\overline{6}
\][/tex]
4. Subtract the original [tex]\(x\)[/tex] from [tex]\(10x\)[/tex] to eliminate the repeating part:
[tex]\[
10x - x = 6.\overline{6} - 0.\overline{6}
\][/tex]
[tex]\[
9x = 6
\][/tex]
5. Solve for [tex]\(x\)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{6}{9}
\][/tex]
6. Simplify the fraction [tex]\(\frac{6}{9}\)[/tex]:
[tex]\[
\frac{6}{9} = \frac{2}{3}
\][/tex]
7. Now, consider the original decimal [tex]\(2.\overline{6}\)[/tex]:
[tex]\[
2.\overline{6} = 2 + 0.\overline{6}
\][/tex]
[tex]\[
2 + 0.\overline{6} = 2 + \frac{2}{3}
\][/tex]
8. Combine the integer part and the fractional part:
[tex]\[
2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3}
\][/tex]
Thus, the simplified fraction for [tex]\(2.\overline{6}\)[/tex] is [tex]\(\frac{8}{3}\)[/tex].
