1. **State the problem:** Divide the mixed numbers $1 \frac{1}{2}$ by $1 \frac{1}{8}$. 2. **Convert mixed numbers to improper fractions:** $1 \frac{1}{2} = \frac{3}{2}$ $1 \frac{1}{8} = \frac{9}{8}$ 3. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{3}{2} \div \frac{9}{8} = \frac{3}{2} \times \frac{8}{9}$$ 4. **Multiply the fractions:** $$\frac{3 \times 8}{2 \times 9} = \frac{24}{18}$$ 5. **Simplify the fraction:** Divide numerator and denominator by their greatest common divisor, which is 6: $$\frac{24 \div 6}{18 \div 6} = \frac{4}{3}$$ 6. **Convert back to a mixed number:** $$\frac{4}{3} = 1 \frac{1}{3}$$ **Final answer:** $$1 \frac{1}{3}$$