Unless I'm missing something, there is a problem with the question as you posted it. As written, the statements do not give us enough information, even when combined so the official answer should not be what you indicated.

Statement 1
Simplify (1) by dividing both sides by a. If a is positive, the inequality is maintained, and (1) becomes x>y. On the other hand, if a is negative, dividing by a will flip the sign, and (1) becomes x<y.
Because we can get conflicting results, statement (1) is NOT SUFFICIENT

Statement 2
We can think of a^2x as (a^2)^x. Likewise, a^2y = (a^2)^y. Thus, (2) can be rewritten as (a^2)^x > (a^2)^y. Now, just think of a^2 as an unknown positive value. If this value is greater than 1, for example if a^2 = 4, then (a^2)^x > (a^2)^y would mean that x>y. However, if this value is less than 1, for example if a^2=1/4, then (a^2)^x > (a^2)^y would mean that x<y (because as the exponent of a proper fraction increases, the result only gets smaller; for example, x=1, y=2 would work).
Because we can get conflicting results, statement (2) is NOT SUFFICIENT

Merge Statements
The quick way: we determined from (1) that x>y if a is positive, but x<y if a is negative. So for the statements together to be sufficient, (2) should give us a way to determine the sign of a. Since (2) gives us info on a^2, there is no way to isolate the sign of a. Thus the statements together are not sufficient..

The other quick way: we determined from (2) that x>y if a^2>1 (meaning if a>1 or <-1), but x<y if a^2<1 (meaning a is between -1 and 1). So for the statements together to be sufficient, (1) should give us a way to determine whether a falls between -1 and 1. Since (1) doesn't put any restriction on the value of a, we cannot determine this. Thus the statements together are not sufficient.

We can prove that the statements are not sufficient together by plugging in values that agree with both statements but still yield conflicting answers.

If a=2, x=2, y=1 both statements are respected, and x>y
If a=-1/2, x=1, y=2 both statements are respected, and x<y
We have conflicting results. The answer is E

If you correctly copied the question, beware of its source.

Hope this helps. Read my signature below for more
-Patrick

Brent@GMATPrepNow wrote:I believe the question should be worded as I have it above.

Makes sense.

To find: x > y

Statement 1:
ax > ay
if a > 0 ==> x > y
if a < 0 ==> x < y
INSUFFICIENT

Statement 2:
a^2x > a^2y
It doesn't matter now whether what sign does "a" has.. as its (a)^2
So, we can strike-out a^2 from both sides
x > y
SUFFICIENT

Answer [spoiler]{B}[/spoiler]

Patrick_GMATFix wrote:Unless I'm missing something, there is a problem with the question as you posted it. As written, the statements do not give us enough information, even when combined so the official answer should not be what you indicated.

Statement 1
Simplify (1) by dividing both sides by a. If a is positive, the inequality is maintained, and (1) becomes x>y. On the other hand, if a is negative, dividing by a will flip the sign, and (1) becomes x<y.
Because we can get conflicting results, statement (1) is NOT SUFFICIENT

Statement 2
We can think of a^2x as (a^2)^x. Likewise, a^2y = (a^2)^y. Thus, (2) can be rewritten as (a^2)^x > (a^2)^y. Now, just think of a^2 as an unknown positive value. If this value is greater than 1, for example if a^2 = 4, then (a^2)^x > (a^2)^y would mean that x>y. However, if this value is less than 1, for example if a^2=1/4, then (a^2)^x > (a^2)^y would mean that x<y (because as the exponent of a proper fraction increases, the result only gets smaller; for example, x=1, y=2 would work).
Because we can get conflicting results, statement (2) is NOT SUFFICIENT

Merge Statements
The quick way: we determined from (1) that x>y if a is positive, but x<y if a is negative. So for the statements together to be sufficient, (2) should give us a way to determine the sign of a. Since (2) gives us info on a^2, there is no way to isolate the sign of a. Thus the statements together are not sufficient..

The other quick way: we determined from (2) that x>y if a^2>1 (meaning if a>1 or <-1), but x<y if a^2<1 (meaning a is between -1 and 1). So for the statements together to be sufficient, (1) should give us a way to determine whether a falls between -1 and 1. Since (1) doesn't put any restriction on the value of a, we cannot determine this. Thus the statements together are not sufficient.

We can prove that the statements are not sufficient together by plugging in values that agree with both statements but still yield conflicting answers.

If a=2, x=2, y=1 both statements are respected, and x>y
If a=-1/2, x=1, y=2 both statements are respected, and x<y
We have conflicting results. The answer is E

If you correctly copied the question, beware of its source.

Hope this helps. Read my signature below for more
-Patrick

Your work in bold is misinterpretation of the situation presented. [spoiler]B[/spoiler] is the correct answer.