jmc

We can simplify each square root first: $\sqrt{338}=\sqrt{2\cdot 169}=13\sqrt{2}$, $\sqrt{288}=\sqrt{2\cdot 144}=12\sqrt{2}$, $\sqrt{150}=\sqrt{6\cdot 25}=5\sqrt{6}$, and $\sqrt{96}=\sqrt{6\cdot 16}=4\sqrt{6}$. Now we can cancel a lot: $$\frac{13\sqrt{2}}{12\sqrt{2}}+\frac{5\sqrt{6}}{4\sqrt{6}}=\frac{13}{12}+\frac{5}{4}=\frac{13+15}{12}=\frac{28}{12}=\boxed{\frac{7}{3}}.$$