I think the answer should be (C)

1. Lets look at the first case:

3x-3y = 1
so : 3(x-y) = 1
so: x-y = 1/3

I can come up with a pair of values that satisfy this equation either ways:

x = 1, y = 2/3 (both x and y are non negative)

or

x = -1/3 and y = -2/3

Both yield the same result in the equation so INSUFFICIENT

2. Lets look at the second case:

x/y < 1

so: x < y

Lets try two cases again......

Let's look at the same figures as above. In the second case x > y , i.e -1/3 is greater than -2/3. We can also have a whole range of positive integers that satisfy this equation.... so INSUFFICIENT...

3. Now lets use both:

first statement says that x - y is positive so if x < y as per statement 2, they both cannot be of the same sign. Hence statement is SUFFICIENT..

so.... Ans (C)

C IMO

IMO E

Statement one says X > Y Not Sufficient

Statement two says X > Y if Y is negative and X < Y if Y I positive - Not Sufficient

Taking both the statement together Y has to be negative. Does X have to be negative too?

Values X = [spoiler]2/9[/spoiler] and Y = [spoiler]-1/9[/spoiler] Satisfy both conditions.
Also Values X = [spoiler]-1/9[/spoiler] and Y = [spoiler]-4/9[/spoiler] satisfy both conditoins.

chufus wrote:I think the answer should be (C)

2. Lets look at the second case:

x/y < 1

so: x < y

Hey Chufus,

This is wrong. You can't mulitply or divide variables in inequalities unless you know the sign of the variable. Look at ripulgupta's explanation.

ripulgupta nailed it, and prodizy made a great point, too.

Here the question is asking if they're BOTH negative. Statement 1 gives us a nice algebraic start, becuase if you factor the 3:

3x - 3y = 1
3(x-y) = 1
(x - y) = 1/3

You get that the difference is 1/3, which could be done via either:

2/3 - 1/3 (both positive ---> NO)
or
-1/3 - (-2/3) (both negative ---> YES)

Statement 2 is clearly not sufficient on its own. 1/2 satisfies statement 2 and gives a clear NO, but -1/-2 also works and gives the answer YES.

Taking them together, we still have the possibility for YES when we use:

x = -1/3 and y = -2/3, because -1/3 / -2/3 is 1/2.

Note that what statement 2 adds is that y must have a greater absolute value if they share the same sign, but that they can have different signs, too.

Perhaps the quickest way to show that they don't have to both be negative is to consider 0 (ever a gamechanging number). If x is 0 and y = -1/3, both statements are satisfied and the answer is NO, as 0 is neither negative nor positive.

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I like this question as a demonstration of your job when picking numbers, which is to "prove insufficiency". The question wants you to see that two positives don't work, because in that case x is always the larger number (to satisfy statement 1) and that means that statement 2 is not satisfied. But you can have one negative and one positive (as ripulgupta had) or x zero and y negative, and then you don't have two negatives. The GMAT here thinks that you'll only try two positives when you're assessing choice C, realize that that doesn't work, and assume that they both have to be negative. But the creative folks win out here - those of us who try like crazy to get the opposite answer (NO, when YES may have come a bit easier) will win out.

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And to prodizy's comment about Chufus' post, whenever you see an inequality in a DS problem you should immediately have your senses heightened - there's a 90-some-% chance that the GMAT is setting you up for that "can't multiply/divide by a negative without changing the sign" rule. Their favorite way to do that is by making you multiply/divide a variable and not a known number; that way they can sneak that negative past you. If you see an inequality, immediately start thinking about the potential for variables to be negative!!!!!

GmatKiss wrote:Are x and y both negative numbers?

(1) 3x - 3y = 1
(2) x/y < 1

Clearly from stmt 1 (x-y =1/3) we cannot determine the signs of x and Y. INSUFFICIENT

Stmt 2 -- x/y < 1

Not this could either mean y>x or anything else (we still dont know the sign of X and Y )

-4/-5 (is still less than 1)

So, statement 2 is also not sufficient.

Combine the 2 statements.

x-y =1/3 and x/y<1

even if we take Y as negative, the solution to first equation can still be postive.

for example 2/3 -1/3 = 1/3 (where X =2/3 and Y = 1/3) or -1/3+2/3 = 1/3 (where x = -1/3 and Y = -2/3)

and in both cases. we have -2 <1 and 1/2 < 1 [Respectively]

therefore we still don't have any solution.

IMO OPTION E

I posted a solution to a very similar problem here:

https://www.beatthegmat.com/are-x-and-y- ... 89906.html