1. **State the problem:** Divide the mixed number $1 \frac{1}{9}$ by the fraction $\frac{5}{6}$. 2. **Convert the mixed number to an improper fraction:** $1 \frac{1}{9} = \frac{9}{9} + \frac{1}{9} = \frac{10}{9}$. 3. **Recall the division rule for fractions:** To divide by a fraction, multiply by its reciprocal. So, $$\frac{10}{9} \div \frac{5}{6} = \frac{10}{9} \times \frac{6}{5}.$$ 4. **Multiply the fractions:** $$\frac{10}{9} \times \frac{6}{5} = \frac{10 \times 6}{9 \times 5} = \frac{60}{45}.$$ 5. **Simplify the fraction:** Find the greatest common divisor (GCD) of 60 and 45, which is 15. $$\frac{60}{45} = \frac{60 \div 15}{45 \div 15} = \frac{4}{3}.$$ 6. **Convert back to a mixed number if desired:** $$\frac{4}{3} = 1 \frac{1}{3}.$$ **Final answer:** $$1 \frac{1}{3}.$$