The problem asks us to evaluate the expression $\frac{6\frac{1}{2} \cdot 9\frac{1}{4}}{216\frac{1}{4}}$.

1. Problem Description

The problem asks us to evaluate the expression $\frac{6\frac{1}{2} \cdot 9\frac{1}{4}}{216\frac{1}{4}}$.

2. Solution Steps

First, convert the mixed numbers to improper fractions.

$6\frac{1}{2} = \frac{6 \cdot 2 + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2}$

$9\frac{1}{4} = \frac{9 \cdot 4 + 1}{4} = \frac{36 + 1}{4} = \frac{37}{4}$

$216\frac{1}{4} = \frac{216 \cdot 4 + 1}{4} = \frac{864 + 1}{4} = \frac{865}{4}$

Now, substitute these values into the expression:

$\frac{\frac{13}{2} \cdot \frac{37}{4}}{\frac{865}{4}} = \frac{\frac{13 \cdot 37}{2 \cdot 4}}{\frac{865}{4}} = \frac{\frac{481}{8}}{\frac{865}{4}} = \frac{481}{8} \div \frac{865}{4} = \frac{481}{8} \cdot \frac{4}{865} = \frac{481 \cdot 4}{8 \cdot 865} = \frac{481}{2 \cdot 865} = \frac{481}{1730}$.

3. Final Answer

$\frac{481}{1730}$