$4 \frac{1}{2} \times 2 \frac{7}{3}$

Step 1 ：Convert each mixed number to an improper fraction. The first mixed number $4 \frac{1}{2}$ becomes $\frac{9}{2}$ and the second mixed number $2 \frac{7}{3}$ becomes $\frac{13}{3}$.

Step 2 ：Multiply the two fractions together to get $\frac{9}{2} \times \frac{13}{3} = \frac{117}{6}$.

Step 3 ：Convert the result back to a mixed number to get $19 \frac{3}{6}$.

Step 4 ：Simplify the fractional part to get $19 \frac{1}{2}$.

Step 5 ：\(\boxed{19 \frac{1}{2}}\) is the final answer.