1. We are asked to convert the repeating decimal $0.13\overline{5}$ into a fraction in its simplest form. 2. Let $x = 0.1355555\dots$ where the digit $5$ repeats indefinitely. 3. Multiply $x$ by 10 to shift the decimal one place: $$10x = 1.355555\dots$$ 4. Multiply $x$ by 1000 to shift the decimal three places (to the right of the repeating part): $$1000x = 135.55555\dots$$ 5. Subtract the first equation from the second to eliminate the repeating part: $$1000x - 10x = 135.55555\dots - 1.355555\dots$$ $$990x = 134.2$$ 6. Solve for $x$: $$x = \frac{134.2}{990}$$ 7. Multiply numerator and denominator by 10 to clear the decimal: $$x = \frac{1342}{9900}$$ 8. Simplify the fraction by dividing numerator and denominator by their greatest common divisor (2): $$x = \frac{671}{4950}$$ 9. Check if it can be simplified further; 671 is prime relative to 4950, so this is the simplest form. **Final answer:** $$\boxed{\frac{671}{4950}}$$