Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 06 Jan 2008
Posts: 211

Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
Updated on: 05 Apr 2018, 13:19
Question Stats:
55% (01:43) correct 45% (01:41) wrong based on 2084 sessions
HideShow timer Statistics
Are x and y both positive? (1) \(2x  2y = 1\) (2) \(\frac{x}{y} > 1\)
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by Manbehindthecurtain on 03 May 2008, 09:11.
Last edited by Bunuel on 05 Apr 2018, 13:19, edited 4 times in total.
Edited the question.




Math Expert
Joined: 02 Sep 2009
Posts: 59587

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
19 Jul 2010, 11:32
9. Inequalities For more check Ultimate GMAT Quantitative Megathread Hope it helps.
_________________



Math Expert
Joined: 02 Sep 2009
Posts: 59587

Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
10 May 2010, 14:07
Are x and y both positive?(1) \(2x2y=1\). Well this one is clearly insufficient. You can do it with number plugging OR consider the following: both x and y are positive means that point (x,y) is in the I quadrant. \(2x2y=1\) > \(y=x\frac{1}{2}\). We know it's an equation of a line and basically the question asks whether this line (all (x,y) points of this line) is only in I quadrant. This line for sure passes I quadrant (for example, x = 1.5 and y = 1) but it cannot entirely be only in I quadrant, so there must be some (x, y) points whose coordinates are not both positive. Not sufficient. (2) \(\frac{x}{y}>1\) > x and y have the same sign. But we don't know whether they are both positive or both negative. Not sufficient. (1)+(2) Again it can be done with different approaches. You should just find the one which is the less timeconsuming and comfortable for you personally. One of the approaches: From (1): \(2x2y=1\), so \(x=y+\frac{1}{2}\) From (2): \(\frac{x}{y}>1\); Substitute x into (2): \(\frac{y + \frac{1}{2}}{y}>1\); \(1 + \frac{1}{2y}>1\); \(\frac{1}{2y}>0\); \(y > 0\). \(y\) is positive. Thus, \(x=y+\frac{1}{2}=positive+positive=positive\). So, \(x\) is positive too. Sufficient. Answer: C. You can also check GRAPHIC APPROACH below.
_________________



Math Expert
Joined: 02 Sep 2009
Posts: 59587

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
05 Apr 2018, 17:11
Are x and y both positive?GRAPHIC APPROACH.Notice that the question is basically asks whether the point (x, y) is in the first quadrant. (1) \(2x  2y = 1\). Draw line \(y=x\frac{1}{2}\): Not sufficient. (2) \(\frac{x}{y} > 1\). Draw line \(\frac{x}{y}=1\). The solutions is the green region: Not sufficient. (1)+(2) Intersection is the portion of the blue line which lies in the first quadrant. Sufficient. Answer: C. Attachment:
graph.png [ 6.13 KiB  Viewed 23071 times ]
Attachment:
graph %282%29.png [ 5.95 KiB  Viewed 23080 times ]
_________________




Manager
Joined: 25 Jun 2012
Posts: 61
Location: India
WE: General Management (Energy and Utilities)

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
16 Aug 2012, 09:35
I got it correct but took approx 2.5 minutes.
stmnt 1 : insufficient by plugging numbers stmnt 2 : x/y >1 => not suff as bot x and y can be ve or both +ve.
combined :
from stmnt 1 we have
x=y+1/2 => x/y = 1+1/2y => suppose x/y is 2 as x/y>1 => 2=1+1/2y = y=1/2 henc x=1
both positive.
Ans C
Am I right in my approach?




Intern
Joined: 06 May 2010
Posts: 5

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
11 May 2010, 04:40
I'm still not clear why X and Y has to be positive when X/Y > 1. Can you please explain the way you combined taking both X and Y to be positive and also X and Y as negative. Since in either case X/Y will be > 1.
Thanks H



Math Expert
Joined: 02 Sep 2009
Posts: 59587

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
11 May 2010, 04:59
harikattamudi wrote: I'm still not clear why X and Y has to be positive when X/Y > 1. Can you please explain the way you combined taking both X and Y to be positive and also X and Y as negative. Since in either case X/Y will be > 1.
Thanks H From (2) \(\frac{x}{y}>1\), we can only deduce that x and y have the same sigh (either both positive or both negative). When we consider two statement together: From (1): \(2x2y=1\) > \(x=y+\frac{1}{2}\) From (2): \(\frac{x}{y}>1\) > \(\frac{x}{y}1>0\) > \(\frac{xy}{y}>0\) > substitute \(x\) from (1) > \(\frac{y+\frac{1}{2}y}{y}>0\)> \(\frac{1}{2y}>0\) ( we can drop 2 as it won't affect anything here and write as I wrote \(\frac{1}{y}>0\), but basically it's the same) > \(\frac{1}{2y}>0\) means \(y\) is positive, and from (2) we know that if y is positive x must also be positive. OR: as \(y\) is positive and as from (1) \(x=y+\frac{1}{2}\), \(x=positive+\frac{1}{2}=positive\), hence \(x\) is positive too. Hope it helps.
_________________



Intern
Joined: 17 Feb 2010
Posts: 6

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
24 Jun 2010, 21:03
Here is my confusion. Here is how I approached the question 1. 2x2y=1 so xy=.5
now x=1, y=.5 or x=1/4, y=1/4 so can't tell
2. x/y>1 x>y so again can't tell.
Now if we combine both still the options x=1, x=.5 is true and so is the option x=1/4, y=1/4 true
So can't tell hence E. I know this is not the correct answer but what am I missing?



Math Expert
Joined: 02 Sep 2009
Posts: 59587

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
25 Jun 2010, 04:18
sam2010 wrote: Here is my confusion. Here is how I approached the question 1. 2x2y=1 so xy=.5
now x=1, y=.5 or x=1/4, y=1/4 so can't tell
2. x/y>1 x>y so again can't tell.
Now if we combine both still the options x=1, x=.5 is true and so is the option x=1/4, y=1/4 true
So can't tell hence E. I know this is not the correct answer but what am I missing? Problem with your solution is that the red part is not correct. \(\frac{x}{y}>1\) does not mean that \(x>y\). If both x and y are positive, then \(x>y\), BUT if both are negative, then \(x<y\). From (2) \(\frac{x}{y}>1\), we can only deduce that x and y have the same sigh (either both positive or both negative).
_________________



Intern
Joined: 16 Jul 2010
Posts: 17

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
20 Jul 2010, 12:41
I found this one easiest to solve by drawing a graph. Clearly 1) and 2) alone are not sufficient as discussed, so what remains to be seen is if 2) adds enough information to 1) to determine if both x and y are positive.
Drawing a quick graph of the line y=x1/2 we find that the xintercept of the line is (0.5,0) and the yintercept is (0,0.5). From this graph we can clearly see that we don't need to worry about anything in the 4th quadrant (+x/y is not >1) or the 3rd quadrant (x<y, therefore x/y is not >1). All that is left is the 1st quadrant, in which x and y are both positive.
Sufficient.



Intern
Joined: 17 Aug 2010
Posts: 48

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
30 Sep 2010, 00:48
Manbehindthecurtain wrote: Are x and Y both positive?
1) 2X2Y = 1 2) (x/y) > 1
I guessed and got it right with a 50/50 guess at the end. What I have done here is this 1) 2x  2y = 1 hence x  y = \frac{1}{2} {Dividing both side by 2} In sufficient 2) [fraction]x > y[/fraction] Alone in sufficient When (1) + (2) We can say that if X is greater than y than xy will yield a positive result. Please correct me if I am wrong



Math Expert
Joined: 02 Sep 2009
Posts: 59587

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
30 Sep 2010, 01:03
zerotoinfinite2006 wrote: Manbehindthecurtain wrote: Are x and Y both positive?
1) 2X2Y = 1 2) (x/y) > 1
I guessed and got it right with a 50/50 guess at the end. What I have done here is this 1) 2x  2y = 1 hence x  y = \frac{1}{2} {Dividing both side by 2} In sufficient 2) x > y Alone in sufficient When (1) + (2) We can say that if X is greater than y than xy will yield a positive result.Please correct me if I am wrong First of all: the question is "are x and Y both positive?" not whether "xy will yield a positive result". Next, the red part is not correct. \(\frac{x}{y}>1\) does not mean that \(x>y\). If both \(x\) and \(y\) are positive, then \(x>y\), BUT if both are negative, then \(x<y\). What you are actually doing when writing \(x>y\) from \(\frac{x}{y}>1\) is multiplying both parts of inequality by \(y\): never multiply (or reduce) an inequality by variable (or by an expression with variable) if you don't know the sign of it or are not certain that variable (or expression with variable) doesn't equal to zero.So from (2) \(\frac{x}{y}>1\), we can only deduce that \(x\) and \(y\) have the same sigh (either both positive or both negative). See the complete solution of this problem in my previous post. Hope it helps.
_________________



Retired Moderator
Joined: 16 Nov 2010
Posts: 1232
Location: United States (IN)
Concentration: Strategy, Technology

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
13 Mar 2011, 19:11
1 is not suff, x = 0, y = 1/2 2 is not suff,x and y can be both ve Combining both : x  y = 1/2 and (x  y)/y > 0 so 1/2/y > 0 => y is +ve and because x  y is +ve, x is +ve as well. So answer is C.
_________________
Formula of Life > Achievement/Potential = k * Happiness (where k is a constant) GMAT Club Premium Membership  big benefits and savings



Manager
Joined: 12 Oct 2011
Posts: 149

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
04 Jan 2012, 00:28
C is the answer. Question: Is x > 0 AND y > 0?
Statement 1: 2x  2y = 1 => 2(x  y) = 1 => x  y = 1/2 This just tells us that the difference is positive. But this can be true for cases when both x and y are positive, and when both x and y are negative. For instance, x = 1.5, y = 1 => x  y = 0.5; also, x = 1, y = 1.5 => x  y = 0.5. Thus, INSUFFICIENT.
Statement 2: x/y > 1 This just tells us that x and y have the same sign. That is, both are positive or both are negative. INSUFFICIENT.
Combining these statements, we can use the same numbers used in Statement 1 to find out that both the cases together do not work for negative numbers. For instance, x = 1, y = 1.5 => x  y = 0.5. However, x/y < 1. This violates statement 2.
Thus, the combination of the given statements tells us that x and y both have to be positive. => x > 0 AND y > 0. SUFFICIENT.



SVP
Status: Top MBA Admissions Consultant
Joined: 24 Jul 2011
Posts: 1916

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
12 May 2012, 01:36
Statement (1): xy = 1/2. We can have x=1,y=1/2. Can also have x=0,y=1/2. Insufficient. Statement (2): x/y>1. We can have x=3,y=2. Can also have x=3,y=2. Insufficient. Combining both, (y+1/2)/y > 1 => 1/2y>0 => y>0 Also as x/y>1, x must be>0. Sufficient. C it is.
_________________
GyanOne [www.gyanone.com] Premium MBA and MiM Admissions Consulting
Awesome Work  Honest Advise  Outstanding Results Reach Out, Lets chat!Email: info at gyanone dot com  +91 98998 31738  Skype: gyanone.services



Manager
Joined: 07 Sep 2011
Posts: 64
GMAT 1: 660 Q41 V40 GMAT 2: 720 Q49 V39
WE: Analyst (Mutual Funds and Brokerage)

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
24 Aug 2012, 13:30
Are x and y both positive?
1) 2x2y=1 2(xy)=1 xy=1/2 >3/41/4=1/2....YES >1/4(3/4)=1/2...NO INSUFFICIENT
2) x/y>1 This just means that x and y have the same sign. They're either both positive or both negative. INSUFFICIENT
1&2) x=1/2+y
(1/2+y)/y>1 y/2 + 1 > 1 y/2 > 0 which means that Y is greater than 0. And since both x and y have the same sign, both x and y are Positive. YES.
Answer is C.



Senior Manager
Joined: 13 Aug 2012
Posts: 398
Concentration: Marketing, Finance
GPA: 3.23

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
17 Jan 2013, 04:55
Manbehindthecurtain wrote: Are x and y both positive?
(1) 2x2y = 1 (2) x/y > 1 1. xy = 1/2 This means that the distance between x and y is 1/2 unit and that x is greater than y. But x and y could be positive such as x=5 and y=4.5, OR x and y could be both negative such as x=4 and y=4.5 INSUFFICIENT. 2. x/y > 1 This shows that x and y must be positive meaning they are either both (+) or both (). ex) x/y = 5/2 OR x/y = 5/2 = 5/2 still > 1 INSUFFICIENT. Combine. Let x = 5 and y=9/2: 5/(9/2) = 10/9 > 1  This means when x and y are both positive it could be a solution to x/y > 1 Let x = 4 and y=9/2: 4/(9/2) = 8/9 < 1  This means when x and y are negative it could not be a solution to x/y > 1 Thus, SUFFICIENT that x and y are both positive. Answer: C



Manager
Joined: 07 Sep 2010
Posts: 246

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
14 Sep 2013, 22:23
Hello Bunuel, Request you to please provide your comments on the doubt posted here
Usually, whenever I see combining an inequality and equation, I substitute the value of one of the variable in the inequality and then analyze the effect. So, going by that approach;
xy=1/2 (1) x/y>1 (2) Substituting the value of x in equation(2)
(y+1/2)/y>1
Lets assume that y is positive
(y+1/2) > y
1/2>0 This means that our assumption is true since 1/2 is greater than Zero. Hence, y > 0
Now, Lets assume that y is negative
Now, here I'm stuck, I know that multiplying by a negative number changes the sign of the inequality. I'm sure that the sign will be changed but what would be the resulting equation. I mean, do we need to replace y with "y" in the whole equation. Please clarify. Which of the following would be correct then
a) y+1/2 <y b) y+1/2 < y c) y+1/2 < y
Please help. Thanks



Verbal Forum Moderator
Joined: 10 Oct 2012
Posts: 583

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
14 Sep 2013, 22:39
imhimanshu wrote: Hello Bunuel, Request you to please provide your comments on the doubt posted here
Usually, whenever I see combining an inequality and equation, I substitute the value of one of the variable in the inequality and then analyze the effect. So, going by that approach;
xy=1/2 (1) x/y>1 (2) Substituting the value of x in equation(2)
(y+1/2)/y>1
Lets assume that y is positive
(y+1/2) > y
1/2>0 This means that our assumption is true since 1/2 is greater than Zero. Hence, y > 0
Now, Lets assume that y is negative
Now, here I'm stuck, I know that multiplying by a negative number changes the sign of the inequality. I'm sure that the sign will be changed but what would be the resulting equation. I mean, do we need to replace y with "y" in the whole equation. Please clarify. Which of the following would be correct then
a) y+1/2 <y b) y+1/2 < y c) y+1/2 < y
Please help. Thanks Refer to the highlighted portion : Actually you don't have to take 2 cases at this point: The expression you have is : \(\frac{y+0.5}{y}>1 \to 1+\frac{0.5}{y}>1 \to \frac{1}{y}>0\)> Hence, y>0. As for your doubt, if y is negative, we crossmultiply it and get : \(y+0.5<y \to 0>0.5\), which is absurd. If y is negative, then y would be positive, and for multiplying a positive quantity, you don't need to flip signs. So , yes expression a is correct.
_________________



Intern
Joined: 23 Oct 2012
Posts: 22

Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
Show Tags
03 Dec 2013, 23:07
St 1) 2x2y = 1 => 2 (xy) = 1 => xy =1/2 => all this tells us is that x > y (could be positive or negative) == hence INSUFF
St 2) x/y > 1 => we don't know if y is (+) or () . So we have two cases:
if y positive, then x>y; if y negative, then x<y (again INSUFF)
Combining 1) and 2) we get x>y (from 1) ...which means y is positive (from 2)
Hence, if y is positive, and x >y, then x is also positive. SUFF!!
Hope this was reasoned properly.




Re: Are x and y both positive? (1) 2x  2y = 1 (2) x/y > 1
[#permalink]
03 Dec 2013, 23:07



Go to page
1 2 3
Next
[ 47 posts ]



