1. Not sufficient

as different values of x yields different y.

2. Not sufficient
x>0 y>0
x<0 y<0

together x/y>1 => (y+1/2)/y >1 =>y >0 so x>0

sufficient. Answer is C.

As quoted - clearly 1 and 2 alone are insufficient.

Another approach for combined:
2x-2y=1
x-y=1/2

x/y>1
if both -ve, then
x < y .....
x - y < 0

if both +ve, then
x > y .....
x - y > 0
This holds good as statement 1 says x - y =1/2

Hence X and Y are +ve, Hence C

For x = -1.5 and y = -2, the ratio x/y = 3/4 < 1, so the condition in (2) isn't fulfilled.
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Here is what I did, and maybe it will help out those not using (or struggling with) the algebra method mentioned earlier in Bunuel's post. It looks long, but I tried to be complete with the thought and reasoning process

1) 2x-2y = 1

First, test numbers:
If x is a positive number, then y must be a positive for the math to work. I chose x=2 to evaluate this, which leads to y = \frac{3}{2}
2(2) - 2(\frac{3}{2}) = 1
Are X and Y both positive? Yes

If x is a negitive number, then y must be a negitive for the math to work. I chose x=-2 to evaluate this, which leads to y= \frac{-5}{2}
2(-2) - 2(\frac{-5}{2}) = 1
Are X and Y both positive? No

From these examples, we can also see that x is always \frac{1}{2} more than y -----> x=y+\frac{1}{2}
We have also learned that both variables must be the same sign

1) is not sufficient



2) \frac{x}{y} > 1

First thing this tells me is that x and y are both the same sign because the result is positive
It also tells me that |x| >|y|, because the absolute numerator MUST be larger than the absolute denominator to be grater than 1.

So far this tells me nothing, but let's throw in some numbers to see what is happening:
If x is a positive number, than y must be positive. I picked x = 4, so y must be a positive number less than 4; I chose y = 2.
\frac{4}{2} = 2 which is greater than 1
Are X and Y both positive? Yes

If x is negitive number, than y must be negitive. I picked x = -4, so y must be a negitive number whose ABSOLUTE value is less than 4. I chose y = -2
\frac{-4}{-2} = 2 which is greater than 1
Are X and Y both positive? No

2) is not sufficient



1+2) We have three conditions established from 1) and 2):
c1 - The signs of x and y must be the same
c2 - The absolute value of x is larger than the absolute value of y
c3 - x is always \frac{1}{2} more than y ----> x=y+\frac{1}{2}

We also have some examples used thus far when evaluating statement 1 by itself, let's see what meets the criteria once the conditions are considered simultaneously:

- Both positive; x = 2 and y = \frac{3}{2}
c1 - pass, signs are the same
c2 - pass, 2 > \frac{3}{2}
c3 - pass, 2 = \frac{3}{2} + \frac{1}{2}
1) 2x-2y=1; yes, 2(2) - 2(\frac{3}{2})=1
2) \frac{x}{y} > 1; yes, \frac{2}{(3/2)} = \frac{4}{3}; \frac{4}{3}>1

- Both negitive; No values will fit this condition when considering both statements:
This idea violates c2 (|x| is not larger than |y|); when a negitive value for x is \frac{1}{2} greater than y, the absolute value of the numerator is always going to be smaller than the denominator , making \frac{x}{y} < 1. Therefore, both being positive is the only option to fit both statements.

Example: x = -2, so y = \frac{-5}{2} as per what we discovered in statement one, x = y+\frac{1}{2}
When applying these values to statement 2, we can see that \frac{x}{y} = \frac{-2}{(-5/2)} = .8< 1


1+2) is sufficient