The problem is to simplify the following expression: $\frac{6\frac{1}{2} \cdot 9\frac{1}{4}}{216\frac{1}{4}}$

1. Problem Description

The problem is to simplify the following 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 the improper fractions into the expression:

$\frac{\frac{13}{2} \cdot \frac{37}{4}}{\frac{865}{4}}$

Simplify the numerator:

$\frac{13}{2} \cdot \frac{37}{4} = \frac{13 \cdot 37}{2 \cdot 4} = \frac{481}{8}$

Now, divide the numerator by the denominator:

$\frac{\frac{481}{8}}{\frac{865}{4}} = \frac{481}{8} \div \frac{865}{4} = \frac{481}{8} \cdot \frac{4}{865} = \frac{481}{2 \cdot 4} \cdot \frac{4}{865} = \frac{481}{2} \cdot \frac{1}{865} = \frac{481}{2 \cdot 865} = \frac{481}{1730}$

Since $865 = 5 \cdot 173$ and $481$ is not divisible by 5 or 173, the fraction is in simplest form.

3. Final Answer

$\frac{481}{1730}$