1. **State the problem:** We need to find the value of $a$ in the equation $$\frac{\sqrt{3}}{5} + \frac{2}{\sqrt{3}} = a\sqrt{3}$$ where $a$ is a fraction. 2. **Rewrite the equation:** The goal is to express the left side as a single term multiplied by $\sqrt{3}$. 3. **Simplify the second term:** To combine terms, rationalize the denominator of $\frac{2}{\sqrt{3}}$: $$\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$$ 4. **Rewrite the equation with the rationalized term:** $$\frac{\sqrt{3}}{5} + \frac{2\sqrt{3}}{3} = a\sqrt{3}$$ 5. **Factor out $\sqrt{3}$ on the left side:** $$\sqrt{3} \left( \frac{1}{5} + \frac{2}{3} \right) = a\sqrt{3}$$ 6. **Simplify inside the parentheses:** Find common denominator 15: $$\frac{1}{5} = \frac{3}{15}, \quad \frac{2}{3} = \frac{10}{15}$$ $$\frac{3}{15} + \frac{10}{15} = \frac{13}{15}$$ 7. **So the equation becomes:** $$\sqrt{3} \times \frac{13}{15} = a\sqrt{3}$$ 8. **Divide both sides by $\sqrt{3}$ to isolate $a$:** $$\cancel{\sqrt{3}} \times \frac{13}{15} = a \cancel{\sqrt{3}}$$ $$a = \frac{13}{15}$$ **Final answer:** $$a = \frac{13}{15}$$