The average of x numbers is y² and the average of y numbers is x². So the average of all the numbers taken together is:

$$\eqalign{ & {\text{According to the question,}} \cr & {\text{Average of }}x{\text{ number}} = {y^2} \cr & \therefore {\text{Sum of }}x{\text{ number}} = x{y^2} \cr & {\text{Average of }}y{\text{ number}} = {x^2} \cr & \therefore {\text{Sum of }}y{\text{ number}} = y{x^2} \cr & {\text{Average of all number}} = \frac{{x{y^2} + y{x^2}}}{{x + y}} \cr & = \frac{{xy\left( {x + y} \right)}}{{x + y}} \cr & = xy \cr} $$