1. **State the problem:** Convert the recurring decimal $0.26\dot{6}$ (where 6 repeats indefinitely) into a fraction in simplest form. 2. **Understand the notation:** The decimal $0.26\dot{6}$ means $0.266666\ldots$ with 6 repeating infinitely. 3. **Set up the equation:** Let $x = 0.2666\ldots$ 4. **Multiply to isolate the repeating part:** Since the repeating digit is one digit long, multiply by 10 to shift one decimal place: $$10x = 2.6666\ldots$$ 5. **Multiply to isolate the non-repeating part:** Multiply by 100 to shift two decimal places (to cover the non-repeating part 0.26): $$100x = 26.6666\ldots$$ 6. **Subtract the two equations to eliminate the repeating decimal:** $$100x - 10x = 26.6666\ldots - 2.6666\ldots$$ $$90x = 24$$ 7. **Solve for $x$:** $$x = \frac{24}{90}$$ 8. **Simplify the fraction:** Divide numerator and denominator by their greatest common divisor, 6: $$\frac{24 \div 6}{90 \div 6} = \frac{4}{15}$$ **Final answer:** $$0.26\dot{6} = \frac{4}{15}$$