To find the simplified fraction that is equal to the repeating decimal \(0.1 \overline{7}\), we follow these steps:
1. Assign Variable: Let \( x \) be equal to the repeating decimal.
[tex]\[
x = 0.177777\ldots
\][/tex]
2. Multiply by Power of 10: Multiply both sides of the equation by 10 to shift the decimal point one place to the right.
[tex]\[
10x = 1.77777\ldots
\][/tex]
3. Subtract Equations: Subtract the first equation from the second to eliminate the repeating part.
[tex]\[
10x - x = 1.77777\ldots - 0.17777\ldots
\][/tex]
Simplifies to:
[tex]\[
9x = 1.6
\][/tex]
4. Solve for x: Isolate \( x \) by dividing both sides by 9.
[tex]\[
x = \frac{1.6}{9}
\][/tex]
5. Convert Decimal to Fraction: Convert \(1.6\) (1.6 is equivalent to \(\frac{16}{10}\)) to a fraction.
[tex]\[
x = \frac{\frac{16}{10}}{9} = \frac{16}{10} \cdot \frac{1}{9} = \frac{16}{90}
\][/tex]
6. Simplify Fraction: Simplify the fraction \(\frac{16}{90}\) by finding the greatest common divisor (GCD) of 16 and 90.
[tex]\[
\text{GCD of 16 and 90 is 2.}
\][/tex]
[tex]\[
\frac{16}{90} = \frac{16 \div 2}{90 \div 2} = \frac{8}{45}
\][/tex]
Therefore, the fraction that is equal to \(0.1 \overline{7}\) is \(\frac{8}{45}\).
The correct answer is:
[tex]\[
\boxed{\frac{8}{45}}
\][/tex]
