1. **State the problem:** Multiply the fractions $\frac{2}{3}$ and $4 \frac{1}{2}$.\n\n2. **Convert mixed number to improper fraction:**\n$4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2}$.\n\n3. **Write the multiplication:**\n$\frac{2}{3} \times \frac{9}{2}$.\n\n4. **Multiply numerators and denominators:**\n$\frac{2 \times 9}{3 \times 2} = \frac{18}{6}$.\n\n5. **Simplify the fraction:**\n$\frac{\cancel{18}^{3 \times 6}}{\cancel{6}^{3 \times 2}} = \frac{3}{1} = 3$.\n\n6. **Final answer:**\n$3$.