Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 15 Mar 2014, 18:55

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# x and y are positive integers. If 1/x + 1/y < 2, which of

Author Message
TAGS:
Manager
Joined: 10 Jul 2009
Posts: 131
Location: Ukraine, Kyiv
Followers: 2

Kudos [?]: 19 [0], given: 60

x and y are positive integers. If 1/x + 1/y < 2, which of [#permalink]  05 Sep 2009, 09:59
00:00

Difficulty:

35% (medium)

Question Stats:

61% (02:14) correct 38% (01:13) wrong based on 126 sessions
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
[Reveal] Spoiler: OA

_________________

Never, never, never give up

Senior Manager
Joined: 02 Aug 2009
Posts: 269
Followers: 3

Kudos [?]: 79 [0], given: 1

Re: X and Y. which is true? [#permalink]  05 Sep 2009, 10:31
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 given1/x +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: 204
Schools: Sloan '14 (S)
Followers: 3

Kudos [?]: 70 [0], given: 16

Re: X and Y. which is true? [#permalink]  05 Sep 2009, 11:20

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
SVP
Joined: 16 Nov 2010
Posts: 1698
Location: United States (IN)
Concentration: Strategy, Technology
Followers: 28

Kudos [?]: 261 [0], given: 36

Re: X and Y. which is true? [#permalink]  14 Feb 2011, 07:36
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.

_________________

Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant)

Manager
Joined: 20 Jul 2011
Posts: 152
GMAT Date: 10-21-2011
Followers: 0

Kudos [?]: 25 [0], given: 15

Re: X and Y. which is true? [#permalink]  05 Sep 2011, 13:16
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) (X-Y)^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

_________________

"The best day of your life is the one on which you decide your life is your own. No apologies or excuses. No one to lean on, rely on, or blame. The gift is yours - it is an amazing journey - and you alone are responsible for the quality of it. This is the day your life really begins." - Bob Moawab

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 4054
Location: Pune, India
Followers: 865

Kudos [?]: 3634 [1] , given: 144

Re: X and Y. which is true? [#permalink]  05 Sep 2011, 22:20
1
KUDOS
Expert's post
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) (X-Y)^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
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options. Veritas Prep Reviews Manager Status: Still Struggling Joined: 02 Nov 2010 Posts: 139 Location: India GMAT Date: 10-15-2011 GPA: 3.71 WE: Information Technology (Computer Software) Followers: 4 Kudos [?]: 6 [1] , given: 8 Re: X and Y. which is true? [#permalink] 07 Sep 2011, 05:05 1 This post received KUDOS 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... _________________ Appreciation in KUDOS please! Knewton Free Test 10/03 - 710 (49/37) Princeton Free Test 10/08 - 610 (44/31) Kaplan Test 1- 10/10 - 630 Veritas Prep- 10/11 - 630 (42/37) MGMAT 1 - 10/12 - 680 (45/34) Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 4054 Location: Pune, India Followers: 865 Kudos [?]: 3634 [0], given: 144 Re: X and Y. which is true? [#permalink] 07 Sep 2011, 06:54 Expert's post 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) (X-Y)^2>0 (2-2)^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 My Blog Save$100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

Manager
Status: Still Struggling
Joined: 02 Nov 2010
Posts: 139
Location: India
GMAT Date: 10-15-2011
GPA: 3.71
WE: Information Technology (Computer Software)
Followers: 4

Kudos [?]: 6 [0], given: 8

Re: X and Y. which is true? [#permalink]  07 Sep 2011, 07:03
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)
(X-Y)^2>0
(2-2)^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)

_________________

Knewton Free Test 10/03 - 710 (49/37)
Princeton Free Test 10/08 - 610 (44/31)
Kaplan Test 1- 10/10 - 630
Veritas Prep- 10/11 - 630 (42/37)
MGMAT 1 - 10/12 - 680 (45/34)

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 4054
Location: Pune, India
Followers: 865

Kudos [?]: 3634 [0], given: 144

Re: X and Y. which is true? [#permalink]  07 Sep 2011, 19:35
Expert's post
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)
(X-Y)^2>0
(2-2)^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
My Blog

Save \$100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

Manager
Status: Still Struggling
Joined: 02 Nov 2010
Posts: 139
Location: India
GMAT Date: 10-15-2011
GPA: 3.71
WE: Information Technology (Computer Software)
Followers: 4

Kudos [?]: 6 [0], given: 8

Re: X and Y. which is true? [#permalink]  07 Sep 2011, 20:38
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.

_________________

Knewton Free Test 10/03 - 710 (49/37)
Princeton Free Test 10/08 - 610 (44/31)
Kaplan Test 1- 10/10 - 630
Veritas Prep- 10/11 - 630 (42/37)
MGMAT 1 - 10/12 - 680 (45/34)

Senior Manager
Joined: 13 Aug 2012
Posts: 465
Concentration: Marketing, Finance
GMAT 1: Q V0
GPA: 3.23
Followers: 14

Kudos [?]: 148 [0], given: 11

Re: X and Y. which is true? [#permalink]  26 Dec 2012, 22:51
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) (X-Y)^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) (2-2)^2 = 0 OUT!

_________________

Impossible is nothing to God.

Re: X and Y. which is true?   [#permalink] 26 Dec 2012, 22:51
Similar topics Replies Last post
Similar
Topics:
Is 1/x < 1/y for integers x,y ? (1) x^2 + y^2 = 25 (2) xy 7 04 Nov 2005, 18:05
If X>1 and Y>1, is X<Y? (1) X?/(XY+X)<1 (2) 8 21 Mar 2006, 10:09
X and Y are positive integers. If 1/x + 1/y < 2, which of 12 22 Nov 2007, 04:34
X and Y are positive integers. If \frac{1}{X} + \frac{1}{Y} 6 02 Apr 2008, 14:48
x, y are positive integers, if 1/x + 1/y <2, which of the 9 12 May 2008, 23:48
Display posts from previous: Sort by