There could be 4 possibilities:

X = +ive, Y = +ive
X = +ive, Y = -ive
X = -ive, Y = +ive
X = -ive, Y = -ive

On looking at both the equation, I guessed that 3 and 2.5 could be best substitute for the +ive & -ive values of X and Y.

Condition 1: 2x - 2y = 1

So on applying it the above situations we get,
(X = 3, Y = 2.5) satisfy the equation.
(X = 3, Y = -2.5) does not satisfy the equation.
(X = -3, Y = 2.5) does satisfy the equation.
(X = -2.5, Y = -3) satisfy the equation.

so by condition 1, we come to know that either both x and y are positive or both are negative. (NOT SUFFICIENT)

Condition 2: x/y > 1

So on applying it the above situations we get,
(X = 3, Y = 2.5) satisfy the equation.
(X = 3, Y = -2.5) does not satisfy the equation.
(X = -3, Y = 2.5) does satisfy the equation.
(X = -3, Y = -2.5) satisfy the equation.

so by condition 2 also, we come to know that either both x and y are positive or both are negative. (NOT SUFFICIENT)

Now lets take both the condition together:-

The last case that is both x and y are negative are contradict to each other in condition 1 and 2, so x and y could not be negative. (Check carefully)

Hence, to satisfy both conditions x and y should be positive only.

Hence answer is (C)

codesnooker wrote: Now lets take both the condition together:-

The last case that is both x and y are negative are contradict to each other in condition 1 and 2, so x and y could not be negative. (Check carefully)

Hence, to satisfy both conditions x and y should be positive only.

Hence answer is (C)

codesnooker, thanks for your input! But I am not getting one thing -
Initially when u took the condition-1 with the examples, u assumed X = -2.5 and Y = -3 - which holds the condition-1.
But when u tried to take the condition-2 in account, u mentioned X = -3 and Y = -2.5 - This assumption obviously will not validate the condition-1.

So why C? Am I missing something?

I switched the values because initially I want to prove that both statements alone true for negative values along with positive values. --> Only to make both statements as NOT SUFFICIENT.

Now, when we taken both statements together, the equations contradicts (oppose) to each other for negative values. It means if keep same negative values for both the statement, one will hold true and other will not. So, it means x and y can never be negative, as for negative values one is true and another is false.

Only if both X and Y are positive, then both the condition satisfied. Hence both statements together are sufficient to depict that both X and Y should be positive numbers (not integers) and alone none is SUFFICIENT.

Hope you get the logic now.

OK! Now got it, codesnooker! Hey is it not a little confusing question?

Yes, initially it confused me for a second, then I closed my eyes and taken a deep breath and looked on the question. I got the logic in next second.

Another approach:

B) X/Y>1 or |x|>|y| insufficient X=-3 and Y=-2
A) 2X-2Y=1 or x-y=1/2 insufficient X=-2 and Y=-2.5

Together: sufficient because for negative values for A to be true |x|<|y| always