Are x and y both positive? 1) 2x - 2y = 1, 2) x/y > 1

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Harish Dorai

Are x and y both positive? 1) 2x - 2y = 1, 2) x/y > 1

Data sufficiency question:

Are x and y both positive?

1) 2x - 2y = 1

2) x/y > 1

I thought the answer choice is (E), but it is not correct as per GMATPrep software.

My reasoning was as follows:

Statement (1) can be simplified as

x - y = 1/2. This is NOT SUFFICIENT.

Statement (2) can be re-written as x > y. This is NOT SUFFICIENT

Combining the above 2 statements and taking an example as shown below.

x = -3/2 and y = -2. In this case x > y as per the second statement and x - y = 1/2 as per the second statement. Similarly I could also have an example that satisfies x and y as positive. So I went ahead with answer choice (E).

However what I failed to realize in the above example was that it actually doesn't satisfy the original condition x/y > 1. Because -3/2 divided by -2 will give 3/4 which is less than 1.

So is it correct to re-write the inequality x/y > 1 to x > y?
GMAT 2007

Harish another approach: -

I agree (1) or (2) are INSUFFICIENT. but consider both of them together

(1) 2x-2y = 1
(2) x/y >1

Rephrasing (1)

x-y = 1/2
x = y + 1/2

Now substituting value of x in (2)

(y+1/2)/y >1

Solving:

1+ 1/2y > 1

so, 1/2y > 0 it means y >0 since x = y+1/2 s0 X >0 as well.

Hope it helps

GMAT 2007
GMAT 2007

So, the answer should be (C). Am I correct?

GMAT 2007
a7lee

For A) 2x - 2y = 1 -----> x - y = 1/2. You can have 0 - (-1/2) = No or You can have +1 - (+1/2) = Yes. So A is insufficient.

For B) x/y > 1 ---> Either X and Y are both + or X and Y are both negative. So B is insufficent. NOTE: |x| > |y|.

For A+B.

Have x and y be positive and make it work with equation A. So +1 - (+1/2) = 1/2 Yes.
.... be negative and make it work with equation B. So -1 - (-y) = 1/2. y = (3/2) which is > |x|. So you will see that two negatives cannot work because it violtates the rule that x/y > 1. So for A+B the answer is yes.
Harish Dorai

Thanks for the explanation. (C) is the right answer.
givemeanid

Statement (2) can be re-written as x > y.

You CANNOT do this. You do not know at this point whether y > 0 or y < 0.
GMAT 2007

givemeanid,

In my approach, with help of (1) & (2) , I have calculated y. Instead of substituting values, the calculation makes it clear that y >0 and so x>0. Hence both are +ve. I feel it was the better approach rather than picking numbers. It helped to stayaway from intricacies of possibility of -ve values.

GMAT 2007
givemeanid

GMAT 2007, your solution is good. I also tend to do the same before using numbers for inequalities!
dbernst
ManhattanGMAT Staff

Posts: 300
Joined: Mon May 09, 2005 9:03 am

Good discussion all. Now that's teamwork!
unique

I did it this way -

1. 2(x-y) =1
x-y = 1/2 clearly insufficient

2. x/y > 1

x>y when y > 0
x<y when y< 0

insufficient

TOGETHER x-y =1/2 means x>y

From 2 x>y when y>0 so x>0 answer C
Guest

GMAT 2007 wrote:
so, 1/2y > 0 it means y >0

Im sure this is pretty remedial for you guys, but can someone really quickly explain how from 1/2y > 0 = y>0 ??

Thanks!
rfernandez
ManhattanGMAT Staff

Posts: 381
Joined: Fri Apr 07, 2006 8:25 am

Im sure this is pretty remedial for you guys, but can someone really quickly explain how from 1/2y > 0 = y>0 ??

Multiply both sides of the inequality by 2. 2 * 1/2y yields y; 2 * 0 yields 0.
albert.chi
Course Students

Posts: 6
Joined: Sun Aug 10, 2008 8:05 am

Re: Are x and y both positive? 1) 2x - 2y = 1, 2) x/y > 1

Can someone help me understand how my 'intuition' is incorrect?

To me:

Statement 1) says that the difference between X and Y is .5, no matter where on the number line they are. Both cases work:
<-------0-------Y(1)---X(1.5)---->
or
<-Y(-2)---X(-1.5)---0------------>

Statement 2) says that X is greater than Y and that they have the same sign (negative or positive)

-
Now taking these together without plugging them into each other like the solution above, isn't all the information I have just saying that:

X is to the right of Y on the number line, they are spaced .5 apart, and they both have the same sign? (Therefore E)

Can someone please help me find the missing piece in my logic to understand why it's C?

Thanks
albert.chi
Course Students

Posts: 6
Joined: Sun Aug 10, 2008 8:05 am

Re: Are x and y both positive? 1) 2x - 2y = 1, 2) x/y > 1

Hi, can someone please answer my question in case it was missed?

Thanks!
Ben Ku
ManhattanGMAT Staff

Posts: 817
Joined: Sat Nov 03, 2007 7:49 pm

Re: Are x and y both positive? 1) 2x - 2y = 1, 2) x/y > 1

Statement 2) says that X is greater than Y and that they have the same sign (negative or positive)

This quote is incorrect. X > Y only if both are positive. For example if x = 3 and y =2, then x/y = 3/2.

However, if both x and y are negative, then x < y. For example, if x = -3 and y = -2, then x / y = 3/2. Here, x < y.

So (2) basically states that x > y > 0 or x < y < 0. (1) states that x > y, so then they must both be positive. Hope that makes sense.
Ben Ku
Instructor
ManhattanGMAT