Saw this on GMAT Prep Test 1:

Note, let "!=" read "not equal".

If x != y, is (x-y)/(x+y) > 1?

1) x>0
2) y<0.

I cross multiplied and had the question rephrased to: Is x-y > x+y. Then I said the x's cancel out and you're left with is -2y>0?

I said the answer was B. My reasoning was if y is negative, then -2*neg will always be positive and thus you can answer whether it is >0.

However, OA:E.

Can someone please explain where I am going wrong. Thanks!

u cant cross multiply bcos x+y maybe negative


two cases arise
if x+y >0
then cross mutliutple with x+y and not change < to > to get
is y<0

if x+y<0
then cross mutliutple with x+y and change < to > to get
y>0

therefore the question boils down to if x+y> 0 is y<0
and if x+y< 0 is y>0

take 1 , that is x>0
does this help no because we cant say anytihin abt y
take 2
that is y<0

does this help no because we cant say anytihin abt x

Now take 1 and 2 i.e. x>0 and y<0
if we add both we cant say anytihn abt x+y

so ans E

the key observation here is that you can't "cross multiply" because you don't know the sign of the denominator x + y.

of course, to understand that, you first have to understand what "cross multiplying" actually IS: it's just multiplication by both denominators at the same time. if you take A/B = C/D, and multiply both sides by BD, you get AD = BC (the result of "cross multiplication").

so when you take (x - y)/(x + y) > 1/1 and "cross multiply", all you're actually doing is multiplying by (x + y). that is not allowed unless the sign of (x + y) is known, which it isn't.

even if (1) and (2) are both true, the sign of (x + y) still isn't known. so at this point you should just dispense with the theory and PICK NUMBERS, with the goal of making the denominator negative one time and positive the next.

x = 1, y = -2: is 3/(-1) > 1? no.
x = 2, y = -1: is 3/1 > 1? yes.
insufficient
ans = e

The way I approach these problems before delving into the nitty gritty is think about what the statement is actually saying..

So in this case because (x-y) / (x+y) > 1 , it means the either x-y > x+y if they are both positive or x-y < x+y if they are both negative. That's the only way I can get them to be greater than 1.

Now looking at the clues you have

(1) X>0 , this tells me nothing about whether x-y or x+y is positive or negative so insufficient

(2) Y<0 , again same reasoning as above

(1) & (2) now i try to see if there are ways in which I can get x-y and x+y to be the same sign, or different signs. This is quite easy to see because picking a y that is greater than x would cause the signs to flip. also if y was less than x, then the signs would stay the same.

therefore insuff.

Ans is E.

excellent number properties observations. if you can come up with something like this within a decent amount of time, then by all means do so.
but remember that half the game here is time management; if you can't come up with this interpretation more or less right away, then you should jump into the number-plugging process without further delay.