Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 10 Jul 2009
Posts: 85
Location: Ukraine, Kyiv

x and y are positive integers. If 1/x + 1/y < 2, which of
[#permalink]
Show Tags
05 Sep 2009, 10:59
Question Stats:
57% (01:54) correct 43% (02:07) wrong based on 125 sessions
HideShow timer Statistics
x and y are positive integers. If 1/x + 1/y < 2, which of the following must be true? (A) x + y > 4 (B) xy>1 (C) x/y + y/x < 1 (D) (x  y)^2 > 0 (E) None of the above
Official Answer and Stats are available only to registered users. Register/ Login.




Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9876
Location: Pune, India

Re: X and Y. which is true?
[#permalink]
Show Tags
05 Sep 2011, 23:20
barakhaiev wrote: X and Y are positive integers. If 1/X + 1/Y < 2, which of the following must be true?
(A) X+Y>4 (B) X*Y>1 (C) X/Y+Y/X<1 (D) (XY)^2>0 (E) None of the above Trying a few values makes us realize that the only relation that holds is (B). But how can we be sure that (B) holds for all acceptable values of X and Y. 1/X + 1/Y < 2 implies (1/X + 1/Y)/2 < 1 A useful property of positive numbers is AM >= GM Arithmetic Mean >= Geometric Mean Say, the numbers are 1/X and 1/Y AM = (1/X + 1/Y)/2 It is given that (1/X + 1/Y)/2 < 1 so we know that AM < 1 GM = \(\sqrt{\frac{1}{X}*\frac{1}{Y}}\) Since GM <= AM, \(\sqrt{\frac{1}{X}*\frac{1}{Y}}\) < 1 \(\frac{1}{XY} < 1\) (Squaring the inequality) \(XY > 1\) (X and Y are positive so the inequality doesn't change)
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >




Math Expert
Joined: 02 Aug 2009
Posts: 8320

x and y are positive integers. If 1/x + 1/y < 2, which of
[#permalink]
Show Tags
05 Sep 2009, 11:31
ans is b...... since x and y are both +ive int, \(x*y>1\)... only exception being x=y=1 as it is not given they are different integers... however it is given \(\frac{1}{x} +\frac{1}{y}<2.\). this cannot be true if x=y=1.... so one or both have to be > 1
_________________



Manager
Joined: 13 Aug 2009
Posts: 120

Re: X and Y. which is true?
[#permalink]
Show Tags
05 Sep 2009, 12:20
Answer is B:
1/X + 1/Y < 2
The maximum value of 1/X is 1 because if X equals any other number greater than one it will be a fraction. The same is true with 1/Y.
So 1/X and 1/Y will always be less than 2 as long as both X and Y are not both equal to one at the same time.
Another way of putting it is:
X*Y>1



Retired Moderator
Joined: 16 Nov 2010
Posts: 1230
Location: United States (IN)
Concentration: Strategy, Technology

Re: X and Y. which is true?
[#permalink]
Show Tags
14 Feb 2011, 08:36
My take on this : From the equation : x + y < 2xy => xy > (x+y)/2 So if x and y are two different positive integers, taking the two least values as 1 and 2, we have x > 1.5 at least. Hence xy > 1. Answer B.
_________________
Formula of Life > Achievement/Potential = k * Happiness (where k is a constant) GMAT Club Premium Membership  big benefits and savings



Manager
Joined: 20 Jul 2011
Posts: 88
GMAT Date: 10212011

Re: X and Y. which is true?
[#permalink]
Show Tags
05 Sep 2011, 14:16
Quote: X and Y are positive integers. If 1/X + 1/Y < 2, which of the following must be true?
(A) X+Y>4 (B) X*Y>1 (C) X/Y+Y/X<1 (D) (XY)^2>0 (E) None of the above Let X=1, 1+1/Y<2 1/Y<1 1<Y Y>1 when X=1, A > yes and no B > yes C> yes and no D> yes and no Answer: B



Manager
Status: Still Struggling
Joined: 03 Nov 2010
Posts: 111
Location: India
GMAT Date: 10152011
GPA: 3.71
WE: Information Technology (Computer Software)

Re: X and Y. which is true?
[#permalink]
Show Tags
07 Sep 2011, 06:05
Guys, can you guide me how D is not true? coz last time i checked, square of any number is greater than 0. Even if x is less than y, still, it's square would me more than 0..unless, x = y...



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9876
Location: Pune, India

Re: X and Y. which is true?
[#permalink]
Show Tags
07 Sep 2011, 07:54
krishnasty wrote: Guys, can you guide me how D is not true? coz last time i checked, square of any number is greater than 0. Even if x is less than y, still, it's square would me more than 0..unless, x = y... Given: 1/X + 1/Y < 2 Say X = 2, Y = 2 These values satisfy the inequality: 1/2 + 1/2 < 2 But they do not satisfy (D) (XY)^2>0 (22)^2 = 0, not greater than 0 Hence (D) must not be true for all values. There are values that satisfy the inequality but does not satisfy (D)
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Manager
Status: Still Struggling
Joined: 03 Nov 2010
Posts: 111
Location: India
GMAT Date: 10152011
GPA: 3.71
WE: Information Technology (Computer Software)

Re: X and Y. which is true?
[#permalink]
Show Tags
07 Sep 2011, 08:03
Karishma, now i need a confirmation on GMAT questions... lets say that if two unknowns are given (like X and Y ), can we assume that these two are equals? I thought if we say x and y, they are implicitly different numbers.. VeritasPrepKarishma wrote: krishnasty wrote: Guys, can you guide me how D is not true? coz last time i checked, square of any number is greater than 0. Even if x is less than y, still, it's square would me more than 0..unless, x = y... Given: 1/X + 1/Y < 2 Say X = 2, Y = 2 These values satisfy the inequality: 1/2 + 1/2 < 2 But they do not satisfy (D) (XY)^2>0 (22)^2 = 0, not greater than 0 Hence (D) must not be true for all values. There are values that satisfy the inequality but does not satisfy (D)



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9876
Location: Pune, India

Re: X and Y. which is true?
[#permalink]
Show Tags
07 Sep 2011, 20:35
krishnasty wrote: Karishma, now i need a confirmation on GMAT questions... lets say that if two unknowns are given (like X and Y ), can we assume that these two are equals? I thought if we say x and y, they are implicitly different numbers.. VeritasPrepKarishma wrote: krishnasty wrote: Guys, can you guide me how D is not true? coz last time i checked, square of any number is greater than 0. Even if x is less than y, still, it's square would me more than 0..unless, x = y... Given: 1/X + 1/Y < 2 Say X = 2, Y = 2 These values satisfy the inequality: 1/2 + 1/2 < 2 But they do not satisfy (D) (XY)^2>0 (22)^2 = 0, not greater than 0 Hence (D) must not be true for all values. There are values that satisfy the inequality but does not satisfy (D) Until and unless they mention 'distinct numbers' or 'X not equal to Y', X and Y can be equal. The equality can be a deal breaker/maker sometimes so you have to make sure that you have analyzed its effects too.
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Manager
Status: Still Struggling
Joined: 03 Nov 2010
Posts: 111
Location: India
GMAT Date: 10152011
GPA: 3.71
WE: Information Technology (Computer Software)

Re: X and Y. which is true?
[#permalink]
Show Tags
07 Sep 2011, 21:38
Thanks Karishma for the information. Quote: Until and unless they mention 'distinct numbers' or 'X not equal to Y', X and Y can be equal. The equality can be a deal breaker/maker sometimes so you have to make sure that you have analyzed its effects too.



Senior Manager
Joined: 13 Aug 2012
Posts: 397
Concentration: Marketing, Finance
GPA: 3.23

Re: X and Y. which is true?
[#permalink]
Show Tags
26 Dec 2012, 23:51
barakhaiev wrote: X and Y are positive integers. If 1/X + 1/Y < 2, which of the following must be true?
(A) X+Y>4 (B) X*Y>1 (C) X/Y+Y/X<1 (D) (XY)^2>0 (E) None of the above Let x=2 Let y=2 \(\frac{1}{2} + \frac{1}{2}< 2\) A) X+Y=4 OUT! B) 2*2 > 1 HOLD! C) 2/2 + 2/2 = 2 < 1 OUT! D) (22)^2 = 0 OUT! Answer: B



SVP
Status: It's near  I can see.
Joined: 13 Apr 2013
Posts: 1702
Location: India
Concentration: International Business, Operations
GPA: 3.01
WE: Engineering (Real Estate)

x and y are positive integers. If 1/x + 1/y < 2, which of
[#permalink]
Show Tags
23 Oct 2015, 03:26
barakhaiev wrote: x and y are positive integers. If 1/x + 1/y < 2, which of the following must be true?
(A) x + y > 4 (B) xy>1 (C) x/y + y/x < 1 (D) (x  y)^2 > 0 (E) None of the above My Solution:
Given : x and y are positive integers
Stem: 1/x+1/y<2x+y<2xy (as x and y are positive we can cross multiply)
So A) x+y>4 Try x=1 & y = 2 (Not true)
B) xy>1 Try x=1 and y = 2 (Always true) ) [Note (x=1 and y =1 is not possible values because with these values stem doesn't holds true]
This is our answer as not more then one correct answer choice is possible but we can try all choices for more clarity:
C) x/y+y/x<1 Try x=1 and y = 2 (Not true)
D) (xy)^2 Try x=2 and y=2 (Not true)
E) Can never be true
Answer is B
_________________
"Do not watch clock; Do what it does. KEEP GOING."



Board of Directors
Joined: 17 Jul 2014
Posts: 2491
Location: United States (IL)
Concentration: Finance, Economics
GPA: 3.92
WE: General Management (Transportation)

Re: x and y are positive integers. If 1/x + 1/y < 2, which of
[#permalink]
Show Tags
12 Feb 2016, 19:06
barakhaiev wrote: x and y are positive integers. If 1/x + 1/y < 2, which of the following must be true?
(A) x + y > 4 (B) xy>1 (C) x/y + y/x < 1 (D) (x  y)^2 > 0 (E) None of the above I thought it is some kind of trap here.. we can rewrite the original as: x+y<2xy A  x=2, y=2 > x+y is not greater than 4, yet 1/2 + 1/2 < 2. so A is out. B  if x and y are both positive integers, xy>1 all the times  looks good. C  x/y +y/x <1 or x^2 + y^2 < xy  which will never be true, if x and y are positive integers. D  x^2 + y^2 > 2xy  suppose x=2 and y=2. 4+4 = 8. 2*2*2=8. 8=8, it's not an inequality. E  since B works, e is out.



Senior SC Moderator
Joined: 22 May 2016
Posts: 3747

X and Y are positive integers. If 1/x + 1/y < 2, which of the followin
[#permalink]
Show Tags
24 Nov 2017, 17:51
jedit wrote: X and Y are positive integers. If 1/x + 1/y < 2, which of the following must be true?
A. X+Y>4 B. XY>1 C. X/Y + Y/X < 1 D. (xy)^2 > 0 E. none \(\frac{1}{x} + \frac{1}{y} < 2\) Let x = 1 and y = 2 \(\frac{1}{1} + \frac{1}{2} = \frac{3}{2}\) \(\frac{3}{2} < 2\)  Those numbers for x and y work MUST be true? A. X+Y>4  NO 1 + 2 = 3, which is not greater than 4 B. XY>1 YES Because x and y are positive integers, there is only one way XY would NOT be greater than 1: if both x and y = 1. Then XY = 1. But x = y = 1 violates the prompt: their reciprocals summed must be less than 2; in that case, they equal 2. This choice must be true. C. X/Y + Y/X < 1  NO \(\frac{1}{2} + \frac{2}{1}=\frac{5}{2}\) \(\frac{5}{2}\) is not less than 1 D. (xy)^2 > 0 NO For this option, let x=y=2. \((22)^2 = 0^2 = 0\) 0 is not greater than 0 E. none  NO  One of the answers, B, must be true. Answer B
_________________
SC Butler has resumed! Get two SC questions to practice, whose links you can find by date, here.Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it's the only thing that ever has  Margaret Mead



NonHuman User
Joined: 09 Sep 2013
Posts: 13739

Re: X and Y are positive integers. If 1/x + 1/y < 2, which of the followin
[#permalink]
Show Tags
20 Apr 2019, 01:18
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________




Re: X and Y are positive integers. If 1/x + 1/y < 2, which of the followin
[#permalink]
20 Apr 2019, 01:18






