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

 It is currently 20 Nov 2019, 22:33 ### 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.  # If xy > 0, is y/x + x/y > 2?

Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 59183
If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

3
8 00:00

Difficulty:   65% (hard)

Question Stats: 58% (02:00) correct 42% (02:19) wrong based on 335 sessions

### HideShow timer Statistics

If $$xy > 0$$, is $$\frac{y}{x} + \frac{x}{y} > 2$$?

(1) $$y = x - 1$$

(2) $$x = 2y$$ This question was provided by Math Revolution for the Game of Timers Competition _________________
Director  P
Joined: 16 Jan 2019
Posts: 500
Location: India
Concentration: General Management
WE: Sales (Other)
If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

8
$$\frac{x^2+y^2}{xy}>2$$

Since xy>0, we can multiply the stem by xy maintaining the same inequality and we know that neither x nor y can be 0

So, $$x^2+y^2>2xy$$ or $$x^2+y^2-2xy>0$$ or $$(x-y)^2>0$$

So we need to find whether $$(x-y)^2>0$$, this will be true when x-y is not equal to 0

(1) $$y=x-1$$

So, $$x-y=1$$ => We know that x-y is non zero

So Sufficient

(2) $$x=2y$$

Or, $$x-y=y$$

Since y is non zero, $$(x-y)^2 > 0$$

Sufficient

##### General Discussion
VP  D
Joined: 20 Jul 2017
Posts: 1089
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

2
xy > 0
--> Both x>0 & y>0 (OR) x<0 & y<0

(1) y = x - 1
Since both x & y are of same sign, Irrespective of their values, x/y + y/x will always be > 2

eg: x = 2, y = 1
--> 2/1 + 1/2 = 2.5 > 2

x = 5, y = 4
--> 5/4 + 4/5 = 1.25 + 0.8 = 2.05 > 2

Sufficient

(2) x = 2y
--> x/y = 2
--> x/y + y/x = 2 + 1/2 = 2.5 > 2

Sufficient

IMO Option D

Pls Hit Kudos if you like the solution
Manager  G
Joined: 22 May 2015
Posts: 126
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

xy > 0 which means x and y should be of same sign

1) y= x- 1

Substituting in the x/y + y/x we get (2x^2-2x+1) / x(x-1) which can be further reduced to (2x-1)(x-1)/x(x-1) = (2x - 1) / x

So is (2x-1) / x > 2?

(2x-1) / x = 2 - (1/x) For all x >0 the equation will always be < 2
And for all x <0 the equation will always be > 2.

Hence 1 is Not Sufficient.

2) x = 2y

Substituting in the equation we get y / 2y + 2y / y = 2.5 [note: x and y are of same sign always as given in prompt ] . Hence the equation is >2

So statement 2 alone is sufficient. Hence option B.
_________________
Consistency is the Key
Manager  G
Joined: 19 Apr 2017
Posts: 168
Concentration: General Management, Sustainability
Schools: ESSEC '22
GPA: 3.9
WE: Operations (Hospitality and Tourism)
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

2
Question Stem
$$\frac{y}{x}+\frac{x}{y}>2$$
$$y^2+x^2>2xy$$
$$y^2+x^2-2xy>0$$
$$(x-y)^2>0$$

From Statement 1 $$x-y=1$$ -> square both sides
$$(x-y)^2=1$$ sufficient

From Statement 2 $$x=2y$$
$$x/y=2$$
$$y/x=1/2$$
$$\frac{x}{y}+\frac{y}{x}=2+\frac{1}{2}$$

Which is greater than 2, hence sufficient

Manager  G
Joined: 11 Feb 2013
Posts: 216
Location: United States (TX)
GMAT 1: 490 Q44 V15 GMAT 2: 690 Q47 V38 GPA: 3.05
WE: Analyst (Commercial Banking)
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

2
Given, xy > 0 means neither x nor y can be ZERO
SVP  P
Joined: 03 Jun 2019
Posts: 1853
Location: India
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

1
If xy>0,
is y/x+x/y>2
(1) y=x−1
(2) x=2y

y/x + x/y = (x^2 + y^2)/xy
=> y/x+ x/y -2 >0
=> (x^2 + y^2 - 2xy)/xy>0
=> (x-y)^2/xy>0
=> xy>0 since (x-y)^2 is always >0 if x<>y and it is given that xy>0
The expression x/y+y/x>2 except when x=y

1. y = x- 1 => x<>y
(x-y)^2 = 1 > 0
Sufficient.

2. y = 2x
(x-y)^2 = x^2 > 0 since x<>0
Sufficient.

Each statement alone is sufficient.

IMO D
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."

Please provide kudos if you like my post. Kudos encourage active discussions.

My GMAT Resources: -

Efficient Learning
All you need to know about GMAT quant

Tele: +91-11-40396815
Mobile : +91-9910661622
E-mail : kinshook.chaturvedi@gmail.com
ISB School Moderator G
Joined: 08 Dec 2013
Posts: 607
Location: India
Concentration: Nonprofit, Sustainability
Schools: ISB '21
GMAT 1: 630 Q47 V30 WE: Operations (Non-Profit and Government)
If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

1
D, each independently sufficient.

x=2y
So, y/x + x/y = {y^2 + x^2}/xy
Substituting x=2y
{y^2 + 4*y^2}/2y^2, cancelling y^2 from numerator and denominator
(1+4)/2 > 2 sufficient

Statement#1
x=y+1, for y=1, x=2
2 + .5 is > 2
Also, for y=.1, x=1.1
11 + 1/11 is always >2
But
for y=-7, x=-6
7/6 + 6/7 >2,sufficient.
_________________
Kindly drop a '+1 Kudos' if you find this post helpful.

GMAT Math Book

-I never wanted what I gave up
I never gave up what I wanted-

Originally posted by LeoN88 on 03 Jul 2019, 08:20.
Last edited by LeoN88 on 03 Jul 2019, 08:38, edited 2 times in total.
Intern  S
Joined: 30 May 2017
Posts: 35
Concentration: Marketing, General Management
GPA: 3.96
WE: Engineering (Health Care)
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

Question stem =

[(x^2+y^2)/ xy] > 2

Using 2- we can easily get 2.5
by substituting y = x-1

using 1- we get
(y^2/y^2+1) +1 that may or may not be >2
hence not sufficient
Intern  S
Joined: 09 Feb 2019
Posts: 20
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

1
Ans- D

1)
Substitute the value of y in the eq(n)

(x-1/x) + (x/x-1)
x cannot be 0 and 1 since denominator cannot be zero

Put x=2, value= 2.5
Put x=-1 or -2 value of eq(n) still >2
So sufficient

2) Substitute the value of x in terms of y

(y/2y)+(2y/y)
gives 2+1/2 which is greater than 2. So sufficient

Therefore Ans- D
CEO  D
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2977
Location: India
GMAT: INSIGHT
Schools: Darden '21
WE: Education (Education)
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

1
Quote:
If xy>0, is y/x+x/y>2?

(1) y=x−1

(2) x=2y

If xy>0, i.e. signs of both x and y are identical
Now, y/x+x/y will be equal to 2 only if both x and y are equal otherwise it will always be greater than 2?
So question becomes

Is x = y?

(1) y=x−1

i.e. x and y are not equal hence SUFFICIENT

(2) x=2y

i.e. x and y are NOT equal hence SUFFICIENT

_________________
Prosper!!!
GMATinsight
Bhoopendra Singh and Dr.Sushma Jha
e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772
Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi
http://www.GMATinsight.com/testimonials.html

ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION
Manager  S
Joined: 18 Sep 2018
Posts: 100
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

1
IMO D

If xy > 0, is y/x + x/y > 2?
Let's reorder the question:
y/x + x/y > 2 => x^2 + y^2 > 2xy (as xy > 0, we can easily multiply both sides by xy) => (x-y)^2 > 0 => x-y > 0 => x>y
Thus the question asks is x>y ?

St1: y=x-1 => if x=2, y=1 (note x or y cannot be 0 as given xy>0)
That means x>y, sufficient

St2: x=2y => if y=1, x=2
That means x>y, sufficient

Posted from my mobile device
Manager  G
Joined: 27 May 2010
Posts: 199
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

Statements 1 and 2 together are sufficient to answer the question.

Hence option C would be the answer.

Posted from my mobile device
_________________
Please give Kudos if you like the post
Manager  S
Joined: 04 Sep 2016
Posts: 66
Location: Germany
GPA: 3
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

1
Either both x and y are negative or positive.

Lets start with second statement. Direct substitution will give us 2.5> 2

Statement 1 is a bit tricky. First simplify what the question is asking. y/x+x/y>2
LHS take common denominator and take it to RHS y^2 + x^2 > 2xy
This is basically simplified to (x - y)^2 > 0

Now replace statement 1 .... x - (x - 1) > 0 which is 1 > 0

_________________
Proud Wildling The three great essentials to achieve anything worthwhile are, first, hard work; second, stick-to-itiveness; third, common sense."
VP  D
Joined: 19 Oct 2018
Posts: 1080
Location: India
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

1
Given
xy>0 implies that x and y have same sign. Either both are +ve or both are -ve.

y/x+x/y>2 It is possible when x and y have sign, and when x and y are distinct real numbers.
We already know that x and y have same sign. If we know that x and y are distinct, we can give our answer.

Statement 1- x=y-1
x and y are distinct, hence statement is sufficient.

Statement 2- x=2y
Again we can see x and y are distinct, statement is sufficient

IMO D
Director  G
Joined: 22 Nov 2018
Posts: 562
Location: India
GMAT 1: 640 Q45 V35 GMAT 2: 660 Q48 V33 Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

Required is (x^2+y^2)/xy>0 when xy>0
(1) y=x−1 ; Insufficient as when substituted we get -2+1/x

(2) x=2y sufficient; 2+1/2 is greater than 2

IMO B
_________________
Give +1 kudos if this answer helps..!!
Intern  B
Joined: 15 Jun 2019
Posts: 32
Location: Kazakhstan
Schools: Carey '21
GPA: 3.93
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

From the xy>0 >> x>0; y>0 or x<0; y<0

1) y=x-1
(x-1)/x+x/(x-1)>2
Not sufficient to answer the question

2) x=2y
y/2y+2y/y>2
1/2+2>2 Sufficient

Senior Manager  P
Joined: 12 Dec 2015
Posts: 439
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

1
If xy>0, is y/x+x/y>2?

(1) y=x−1 ==>x-y = 1 ; (x-y)^2 = 1 ==> x^2 -2xy + y^2 = 1 > 0 ==> x^2+y^2 > 2xy => (x^2 + y^2)/xy > 2 (because xy > 0) ==> y/x + x/y > 2 --> yes
(2) x=2y ==> x/y = 2 & y/x = 1/2, so y/x+x/y = 2.5 > 2 --> yes

Manager  S
Joined: 08 Jan 2018
Posts: 98
Location: India
GPA: 4
WE: Information Technology (Computer Software)
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

Hi,
Given xy>0 means either x> 0, y>0 or x<0, y<0.
Now, the question:
$$\frac{(y)}{(x)}$$ + $$\frac{(x)}{(y)}$$ >2

=> ($$x^2$$ + $$y^2$$ -2*x*y) >0
=> $$(x-y)^2$$ >0
=> (x-y) >0
=> x > y ?

1. y = x-1
=> x = y+1
if x> 0 , y>0 then x > y.
if x<0 , y<0 then also x>y.
Sufficient.

2. x =2y
if x>0, y>0 then x>y.
if x<0, y<0 then x<y.
Insufficient.

Please hit kudos if you like the solution.
Director  V
Status: Manager
Joined: 27 Oct 2018
Posts: 726
Location: Egypt
GPA: 3.67
WE: Pharmaceuticals (Health Care)
Re: If xy > 0, is y/x + x/y > 2?  [#permalink]

### Show Tags

1
the given says that x,y ≠ zero, and have the same sign

the question can be rephrased to:
is $$\frac{y}{x} + \frac{x}{y} > 2$$
$$\frac{{y^2 + x^2}}{{xy}} > 2$$
$$y^2 + x^2 > 2xy$$
$$x^2 - 2xy + y^2 > 0$$
$$(x-y)^2 > 0?$$
is $$x≠y$$??

statement 1 says that (x-y) = 1, so yes $$(x-y)^2 > 0$$ --> sufficient
statement 2 says that x-y = y, so $$(x-y)^2 > 0$$ --> $$(y)^2 > 0$$ which is correct as y ≠ zero --> sufficient
D
_________________
Thanks for Kudos Re: If xy > 0, is y/x + x/y > 2?   [#permalink] 03 Jul 2019, 08:44

Go to page    1   2   3   4    Next  [ 76 posts ]

Display posts from previous: Sort by

# If xy > 0, is y/x + x/y > 2?  