To rewrite the repeating decimal [tex]\(2.\overline{67}\)[/tex] as a simplified fraction, let's follow these steps:
1. Let [tex]\( x \)[/tex] be the repeating decimal:
[tex]\[
x = 2.\overline{67}
\][/tex]
2. Multiply [tex]\( x \)[/tex] by a power of 10 that matches the length of the repeating part. In this case, the repeating part "67" has a length of 2 digits, so multiply by [tex]\( 10^2 = 100 \)[/tex]:
[tex]\[
100x = 267.\overline{67}
\][/tex]
3. Set up an equation to eliminate the repeating decimal when subtracting. We now have:
[tex]\[
100x = 267.676767... \quad \text{and} \quad x = 2.676767...
\][/tex]
4. Subtract the original equation from this new equation:
[tex]\[
100x - x = 267.676767... - 2.676767...
\][/tex]
[tex]\[
99x = 265
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{265}{99}
\][/tex]
6. Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 265 and 99 is 1. Thus, the fraction is already in its simplest form:
[tex]\[
\frac{265}{99}
\][/tex]
So, the simplified fraction of the repeating decimal [tex]\(2.\overline{67}\)[/tex] is:
[tex]\[
\boxed{\frac{265}{99}}
\][/tex]
