The problem is asking to find the value of the expression $4 \frac{1}{2} \div \frac{4}{5}$ and write the answer as a mixed number.

Step 1: Convert the mixed number to an improper fraction

$4 \frac{1}{2} = \frac{9}{2}$

Step 2: Set up the division of fractions

The expression becomes: $\frac{9}{2} \div \frac{4}{5}$

Step 3: Convert the division into multiplication by the reciprocal

$\frac{9}{2} \times \frac{5}{4} = \frac{9 \cdot 5}{2 \cdot 4} = \frac{45}{8}$

Step 4: Convert the improper fraction to a mixed number

$\frac{45}{8} = 5 \frac{5}{8}$

The answer is $5 \frac{5}{8}$.

Would you like more details on this solution or have any other questions?

Here are five related questions to expand your understanding:

1. How do you convert a mixed number into an improper fraction?
2. What is the reciprocal of a fraction, and why is it used in division?
3. How can you convert an improper fraction back into a mixed number?
4. What are some common mistakes when dividing fractions?
5. How would you multiply two mixed numbers directly?

Tip: When dividing fractions, always remember to multiply by the reciprocal of the divisor.