GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 14 Dec 2019, 06:06 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # X and Y are positive integers. If 1/x + 1/y < 2, which of the followin

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

26 00:00

Difficulty:   55% (hard)

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
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9876
Location: Pune, India
Re: X and Y. which is true?  [#permalink]

### Show Tags

8
6
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

##### General Discussion
Math Expert V
Joined: 02 Aug 2009
Posts: 8320
x and y are positive integers. If 1/x + 1/y < 2, which of  [#permalink]

### Show Tags

2
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
Schools: Sloan '14 (S)
Re: X and Y. which is true?  [#permalink]

### Show Tags

1
1

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 B
Joined: 16 Nov 2010
Posts: 1230
Location: United States (IN)
Concentration: Strategy, Technology
Re: X and Y. which is true?  [#permalink]

### Show Tags

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)

GMAT Club Premium Membership - big benefits and savings
Manager  Joined: 20 Jul 2011
Posts: 88
GMAT Date: 10-21-2011
Re: X and Y. which is true?  [#permalink]

### Show Tags

1
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

Manager  Status: Still Struggling
Joined: 03 Nov 2010
Posts: 111
Location: India
GMAT Date: 10-15-2011
GPA: 3.71
WE: Information Technology (Computer Software)
Re: X and Y. which is true?  [#permalink]

### Show Tags

2
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 V
Joined: 16 Oct 2010
Posts: 9876
Location: Pune, India
Re: X and Y. which is true?  [#permalink]

### Show Tags

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

Manager  Status: Still Struggling
Joined: 03 Nov 2010
Posts: 111
Location: India
GMAT Date: 10-15-2011
GPA: 3.71
WE: Information Technology (Computer Software)
Re: X and Y. which is true?  [#permalink]

### Show Tags

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)
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9876
Location: Pune, India
Re: X and Y. which is true?  [#permalink]

### Show Tags

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

Manager  Status: Still Struggling
Joined: 03 Nov 2010
Posts: 111
Location: India
GMAT Date: 10-15-2011
GPA: 3.71
WE: Information Technology (Computer Software)
Re: X and Y. which is true?  [#permalink]

### Show Tags

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

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!

SVP  V
Status: It's near - I can see.
Joined: 13 Apr 2013
Posts: 1702
Location: India
GPA: 3.01
WE: Engineering (Real Estate)
x and y are positive integers. If 1/x + 1/y < 2, which of  [#permalink]

### Show Tags

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<2---x+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) (x-y)^2 Try x=2 and y=2 (Not true)

E) Can never be true

_________________
"Do not watch clock; Do what it does. KEEP GOING."
Board of Directors P
Joined: 17 Jul 2014
Posts: 2491
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 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

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 V
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

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. (x-y)^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. (x-y)^2 > 0 NO
For this option, let x=y=2.
$$(2-2)^2 = 0^2 = 0$$
0 is not greater than 0

E. none - NO - One of the answers, B, must be true.

_________________
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
Non-Human 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

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
Display posts from previous: Sort by

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