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

Step 1 ：Convert the mixed number $4 \frac{1}{2}$ to an improper fraction. The numerator of the improper fraction is $4 \times 2 + 1 = 9$, and the denominator is $2$. So, $4 \frac{1}{2}$ is equivalent to $\frac{9}{2}$.

Step 2 ：Next, multiply the two fractions together. The product of the numerators is $9 \times 1 = 9$, and the product of the denominators is $2 \times 4 = 8$. So, $\frac{9}{2} \times \frac{1}{4} = \frac{9}{8}$.

Step 3 ：Finally, convert the improper fraction $\frac{9}{8}$ back to a mixed number. The whole number is $9 \div 8 = 1$, and the remainder is $1$. So, $\frac{9}{8}$ is equivalent to $1 \frac{1}{8}$.

Step 4 ：\(\boxed{1 \frac{1}{8}}\) is the final answer.