Irrationality proof and finding another irrational

Suppose $\frac{3+\sqrt{3}}{5} = r$, where $r$ is rational. Then $3 + \sqrt{3} = 5r \implies \sqrt{3} = 5r - 3$. Since $r$ is rational, $5r - 3$ is rational, which contradicts that $\sqrt{3}$ is irrational. Thus $\frac{3+\sqrt{3}}{5}$ is irrational. For the second part, choose an irrational number like $\sqrt{3}$. The sum $\frac{3+\sqrt{3}}{5} + \sqrt{3} = \frac{3 + 6\sqrt{3}}{5}$, which is also irrational.