Mock AIME 5 Pre 2005 Problems/Problem 8

Square both sides to give $x^2+2x\sqrt{y}+y=22+2\sqrt{96}$ . Thus, we have the system of equations $x^2+y=22$ $x^2y=96$ . Thus, we have $y(22-y)=96$ , $y^2-22y+96=0$ , $(y-6)(y-16)=0$ . Thus, we have $y=6$ as $x$ is an integer, and so $x=4$ . Thus, we have $\frac{x}{y}=\frac{4}{6}=\frac{2}{3}$ , so the answer is $2+3=\boxed{5}$ . ~AbbyWong