The problem asks to evaluate the expression $\frac{\frac{1}{3} \times \frac{1}{5} \div \frac{2}{3} \times \frac{1}{4}}{\frac{1}{4} \div \frac{1}{2}}$.

1. Problem Description

The problem asks to evaluate the expression $\frac{\frac{1}{3} \times \frac{1}{5} \div \frac{2}{3} \times \frac{1}{4}}{\frac{1}{4} \div \frac{1}{2}}$.

2. Solution Steps

First, let's simplify the numerator:

$\frac{1}{3} \times \frac{1}{5} \div \frac{2}{3} \times \frac{1}{4}$

Multiplication and division are performed from left to right.

$\frac{1}{3} \times \frac{1}{5} = \frac{1}{15}$

So, $\frac{1}{15} \div \frac{2}{3} \times \frac{1}{4}$.

To divide fractions, we multiply by the reciprocal.

$\frac{1}{15} \div \frac{2}{3} = \frac{1}{15} \times \frac{3}{2} = \frac{3}{30} = \frac{1}{10}$.

Now, $\frac{1}{10} \times \frac{1}{4} = \frac{1}{40}$.

The numerator simplifies to $\frac{1}{40}$.

Next, let's simplify the denominator:

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

To divide fractions, we multiply by the reciprocal.

$\frac{1}{4} \div \frac{1}{2} = \frac{1}{4} \times \frac{2}{1} = \frac{2}{4} = \frac{1}{2}$.

So, the expression becomes $\frac{\frac{1}{40}}{\frac{1}{2}}$.

To divide fractions, we multiply by the reciprocal.

$\frac{1}{40} \div \frac{1}{2} = \frac{1}{40} \times \frac{2}{1} = \frac{2}{40} = \frac{1}{20}$.

3. Final Answer

The final answer is $\frac{1}{20}$