Whether two quantities are greater, smaller or not possible to determine (GRE Quant)? The question is taken from a general GRE math quantitative comparison question. 
Problem: $x>y$ and $xy\ne0$
Quantity A:
$ \displaystyle\frac{x^2}{{1+ \frac{1}{y}} }$
Quantity B:
$  \displaystyle\frac{y^2}{{1+ \frac{1}{x}} }$
How can I definitivly tell without plotting graphs whether Quantity A or B is greater or smaller or different in different ranges, for all Real values? 
 A: After some implications you should examine the below two parts
\begin{align}
 x(x+1) ? y(y+1) 
\end{align}
x > y and (x+1)>(y+1) for each x,y > 0. Then Quantity A > Quantity B
However, this cannot be always valid if x,y < 0 or x>0 and y<0.
The answer to your question is that you cannot define a relation between A and B for all Real values nor different ranges. 
A: For this kind of GRE quant questions, the best thing to do is plug-in numbers.
For $x=2$ and $y=1$,
$A=3$ and $B=\dfrac{3}{2}$; so Quantity A is greater than Quantity B. 
However, if $x=1$ and $y=-1$; Quantity A takes $0$ in the denominator, which is undefined. 
So the relationship cannot be determined from the information given. 
A: I would simplify the expression:
$$\frac{x^2}{1+\frac{1}{y}}?\frac{y^2}{1+\frac{1}{x}}\to \frac{x^2y}{y+1}?\frac{y^2x}{x+1}\to \frac{x}{y+1}?\frac{y}{x+1}\to (x^2+x)?(y^2+y)$$
For positive x and y, it is trivially true that $(x^2+x)>(y^2+y)$ since x>y
For negative x and y, we can quickly find an example where the opposite is true. $(-1)^2+-1=0<2=(-2)^2+-2$ 
This shows that they are different for different ranges.
