1. **Problem:** Evaluate $4 \frac{1}{2} \div \frac{3}{8}$. 2. **Convert mixed number to improper fraction:** $$4 \frac{1}{2} = \frac{9}{2}$$ 3. **Division of fractions rule:** $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ 4. **Apply the rule:** $$\frac{9}{2} \div \frac{3}{8} = \frac{9}{2} \times \frac{8}{3}$$ 5. **Multiply numerators and denominators:** $$\frac{9 \times 8}{2 \times 3} = \frac{72}{6}$$ 6. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{72}^{12} \times 6}{\cancel{6}^1} = 12$$ 7. **Final answer:** $$4 \frac{1}{2} \div \frac{3}{8} = 12$$