1. **State the problem:** Divide the mixed numbers $1 \frac{1}{5}$ by $1 \frac{1}{2}$. 2. **Convert mixed numbers to improper fractions:** $1 \frac{1}{5} = \frac{6}{5}$ because $1 \times 5 + 1 = 6$. $1 \frac{1}{2} = \frac{3}{2}$ because $1 \times 2 + 1 = 3$. 3. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $\frac{6}{5} \div \frac{3}{2} = \frac{6}{5} \times \frac{2}{3}$. 4. **Multiply the fractions:** $\frac{6}{5} \times \frac{2}{3} = \frac{6 \times 2}{5 \times 3} = \frac{12}{15}$. 5. **Simplify the fraction:** $\frac{12}{15} = \frac{4}{5}$ by dividing numerator and denominator by 3. 6. **Final answer:** $1 \frac{1}{5} \div 1 \frac{1}{2} = \frac{4}{5}$.