Is it C - both combined sufficient?

Statement (1) tells us x-y = 1/2. This can occur with or without x and y both being positive. For example, 5 - 4.5 = 1/2 and -4 - -4.5 = 1/2. So, we know (1) alone is not sufficient.

Statement (2) tells us that x and y are either both positive or both negative (for x/y to be positive). If they are both positive, then x must be greater than y (for x/y > 1). Similarly, if they are both negative, then x must be less than y. For example, 5/4 > 1, or -5/-4 > 1. Therefore, (2) alone is not sufficient either.

Combining (1) and (2), we know that x and y must be both positive with x>y or both negative with x<y AND that x - y = 1/2. However, if x and y are both negative with x<y, then x-y will always be negative and cannot be equal to 1/2 (for example -5 - -4 = -1). However, if they are both positive with x>y, then x-y=1/2 can be satisfied. Therefore, we can conclude that x and y are both positive.

(1) x-y=0.5
if x is positive and greater 0.5, y has to be positive too
if x is positive and 0.5, y has to be zero
if x is positive but less 0.5, y has to be negative
if x is zero or less, y has to be negative
NOT sufficient

(2) x/y>1
either x and y are both positive or negative
NOT sufficient

(1) and (2)
if x is positive and greater 0.5, y has to be positive too:
for example: x=0.7, y=0.2: 0.7-0.2=0.5, 0.7/0.2>1

if x < 0, y has to be negative:
here we can't find a combination, where x > y, because y = x - 0.5, thus x and y have to be positive.

C

That's it. C is correct. Tks a lot.

1. x - y = 0.5
INSUFFICIENT.
2. x/y > 1
INSUFFICIENT

Together,
(y+0.5)/y > 1
1 + 1/2y > 1
1/2y > 0
y > 0
x > 0
SUFFICIENT.

@givemeanid: plugging in - great approach