We are given that $\frac{\sqrt{3} + \sqrt{5}}{\sqrt{5}} = x + y\sqrt{15}$, and we need to find the value of $x + y$.

1. Problem Description

We are given that $\frac{\sqrt{3} + \sqrt{5}}{\sqrt{5}} = x + y\sqrt{15}$, and we need to find the value of $x + y$.

2. Solution Steps

First, we simplify the left-hand side of the equation by splitting the fraction:

$$\frac{\sqrt{3} + \sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3}}{\sqrt{5}} + \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3}}{\sqrt{5}} + 1$$

Now, we rationalize the denominator of the first term:

$$\frac{\sqrt{3}}{\sqrt{5}} = \frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{3} \cdot \sqrt{5}}{5} = \frac{\sqrt{15}}{5}$$

So, we have:

$$\frac{\sqrt{3} + \sqrt{5}}{\sqrt{5}} = \frac{\sqrt{15}}{5} + 1 = 1 + \frac{1}{5}\sqrt{15}$$

Comparing this with $x + y\sqrt{15}$, we have $x = 1$ and $y = \frac{1}{5}$.

Therefore, $x + y = 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5}$.

Converting this improper fraction to a mixed number gives $1\frac{1}{5}$.

3. Final Answer

$1\frac{1}{5}$

The answer is C.