We are asked to evaluate the expression $(\frac{1}{5} + \frac{2}{3}) : \sqrt{\frac{4}{25}}$.

1. Problem Description

We are asked to evaluate the expression $(\frac{1}{5} + \frac{2}{3}) : \sqrt{\frac{4}{25}}$.

2. Solution Steps

First, we add the fractions inside the parenthesis. The least common denominator of $5$ and $3$ is $15$. Thus, we have

$$$$

\frac{1}{5} + \frac{2}{3} = \frac{1 \times 3}{5 \times 3} + \frac{2 \times 5}{3 \times 5} = \frac{3}{15} + \frac{10}{15} = \frac{3+10}{15} = \frac{13}{15}.

$$$$

Next, we evaluate the square root:

$$$$

\sqrt{\frac{4}{25}} = \frac{\sqrt{4}}{\sqrt{25}} = \frac{2}{5}.

$$$$

Now, we divide the result of the sum by the result of the square root:

$$$$

\frac{13}{15} : \frac{2}{5} = \frac{13}{15} \div \frac{2}{5} = \frac{13}{15} \times \frac{5}{2} = \frac{13 \times 5}{15 \times 2} = \frac{13 \times 5}{3 \times 5 \times 2} = \frac{13}{3 \times 2} = \frac{13}{6}.

$$$$

3. Final Answer

$\frac{13}{6}$