If x and y are positive, which of the following must be : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 27 Feb 2017, 17:38

### 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

# If x and y are positive, which of the following must be

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 05 Oct 2008
Posts: 274
Followers: 3

Kudos [?]: 402 [4] , given: 22

If x and y are positive, which of the following must be [#permalink]

### Show Tags

13 Oct 2009, 11:10
4
KUDOS
27
This post was
BOOKMARKED
00:00

Difficulty:

75% (hard)

Question Stats:

55% (02:30) correct 45% (01:45) wrong based on 1051 sessions

### HideShow timer Statistics

If x and y are positive, which of the following must be greater than $$\frac{1}{\sqrt{x+y}}$$?

1. $$\frac{\sqrt{x+y}}{2x}$$

2. $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$

3. $$\frac{\sqrt{x}-\sqrt{y}}{x+y}$$

(A) None
(B) 1 only
(C) 2 only
(D) 1 and 3 only
(E) 2 and 3 only
[Reveal] Spoiler: OA

Last edited by Bunuel on 05 Feb 2012, 00:43, edited 1 time in total.
Edited the question and added the OA
SVP
Joined: 30 Apr 2008
Posts: 1887
Location: Oklahoma City
Schools: Hard Knocks
Followers: 40

Kudos [?]: 580 [1] , given: 32

### Show Tags

13 Oct 2009, 11:32
1
KUDOS
C

The easiest way to solve this problem is by picking some numbers for x and y and then solving and comparing. The problem you really have is trying to compare with radicals in the solutions which are not easy to compare.

First we're given $$\frac{1}{sqrt{x-y}}$$ and asked which of the following I, II, or III MUST be larger than $$\frac{1}{sqrt{x-y}}$$.

I. $$\frac{sqrt{x+y}}{2x}$$

II. $$\frac{sqrt{x} + sqrt{y}}{x+y}$$

III. $$\frac{sqrt{x} - sqrt{y}}{x+y}$$

For my numbers, I chose X = 3 and y = 1

You get $$\frac{1}{sqrt{3+1}} = \frac{1}{2}$$

Then for I, II, and III you get:

I. => $$\frac{sqrt{3+1}}{6} = \frac{2}{6} = \frac{1}{3}$$

II. => $$\frac{sqrt{2} + sqrt{1}}{3 + 1} = \frac{sqrt{2} + 1}{4}$$

III. => $$\frac{sqrt{2} - sqrt{1}}{3 + 1} = \frac{sqrt{2} + 1}{4}$$

Now we have to evaluate these numbers.

I. 1/3 is smaller than 1/2, so it does not satisfy the question of which MUST be larger.

II. Square root of 2 + 1 over 4. Even without knowing what the exact number is, we know $$\sqrt{2}$$ is over 1, so add 1 to that and we get something larger than 2 over 4, which will be greater than 1/2. Can't rules out II as the answer yet. Due to the answer choices, we know the answer should be either C or E.

III. Sqrt of 2 minus 1 over 4 we know will be something under 2 over 4, so that's less than half. III cannot work either.

Something to keep in mind is that the question stem does not rule out fractions for possible values of X and Y, but due to the answers, we are able to eliminate III as a possibility and that leaves only C for the answer.

If you got to a point where both II and III was possible, you would need to also pick some fractions for values of X and Y and evaluate. It's not a really quick way to do this, but it will work and also remember that doing these problems when you're used to them is much faster than reading one of my explanations on how to do it.
_________________

------------------------------------
J Allen Morris
**I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$. GMAT Club Premium Membership - big benefits and savings Math Expert Joined: 02 Sep 2009 Posts: 37144 Followers: 7273 Kudos [?]: 96806 [25] , given: 10786 Re: GMATPrep No prop [#permalink] ### Show Tags 13 Oct 2009, 13:30 25 This post received KUDOS Expert's post 20 This post was BOOKMARKED If x and y are positive, which of the following must be greater than $$\frac{1}{\sqrt{x+y}}$$? 1. $$\frac{\sqrt{x+y}}{2x}$$ 2. $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$ 3. $$\frac{\sqrt{x}-\sqrt{y}}{x+y}$$ (A) None (B) 1 only (C) 2 only (D) 1 and 3 only (E) 2 and 3 only First of all $$\frac{1}{\sqrt{x+y}}$$ is always positive. This by the way eliminates option III right away as $$\sqrt{x}-\sqrt{y}$$ (numereator) may or may not be positive, so we should concentrate on I and II Next: $$\sqrt{x}+\sqrt{y}$$ is always great than $$\sqrt{x+y}$$ (well in fact if both $$x$$ and $$y$$ are 0, they are equal but it's not the case as given that $$x$$ and $$y$$ are positive). To check this: square them $$(\sqrt{x}+\sqrt{y})^2=x+2\sqrt{xy}+y>x+y=\sqrt{x+y}^2$$ Let's proceed: SOLUTION #1 $$\frac{1}{\sqrt{x+y}}=\frac{\sqrt{x+y}}{x+y}$$ I. $$\frac{\sqrt{x+y}}{2x}$$ --> nominators are the same, obviously denominator $$2x$$ may or may not be greater than $$x+y$$. OUT. II. $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$ --> denominators are the same and nominator $$\sqrt{x}+\sqrt{y}$$ (as we've already discussed above) is always greater than $$\sqrt{x+y}$$. OK III. Well we can not even consider this one as our expression $$\frac{1}{\sqrt{x+y}}$$ is always positive and the $$\sqrt{x}-\sqrt{y}$$ (numerator) can be negative. OUT Answer C. SOLUTION #2 The method called cross multiplication: Suppose we want to know which positive fraction is greater $$\frac{9}{11}$$ or $$\frac{13}{15}$$: crossmultiply $$9*15=135$$ and $$11*13=143$$ --> $$135<143$$ which fraction gave us numerator for bigger value 143? $$\frac{13}{15}$$! Thus $$\frac{13}{15}>\frac{9}{11}$$. Lets do the same with our problem: I. $$\frac{\sqrt{x+y}}{2x}$$ and $$\frac{1}{\sqrt{x+y}}$$ --> $$\sqrt{x+y}*\sqrt{x+y}=x+y$$ and $$2x*1=2x$$. $$x+y$$ may or may not be greater than $$2x$$. OUT II. $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$ and $$\frac{1}{\sqrt{x+y}}$$ --> $$(\sqrt{x}+\sqrt{y})(\sqrt{x+y})$$ and $$x+y$$. Divide both sides by $$\sqrt{x+y}$$ --> $$\sqrt{x}+\sqrt{y}$$ and $$\sqrt{x+y}$$. We know that $$\sqrt{x}+\sqrt{y}$$ is always greater, which one gave the numerator for it: $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$, so $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$ is always greater than $$\frac{1}{\sqrt{x+y}}$$. OK III. Well we can not even consider this one as our expression $$\frac{1}{\sqrt{x+y}}$$ is always positive and the $$\sqrt{x}-\sqrt{y}$$ (numerator) can be negative. OUT Answer C. Hope it's clear. _________________ SVP Joined: 30 Apr 2008 Posts: 1887 Location: Oklahoma City Schools: Hard Knocks Followers: 40 Kudos [?]: 580 [4] , given: 32 Re: GMATPrep No prop [#permalink] ### Show Tags 13 Oct 2009, 13:39 4 This post received KUDOS Those are good methods and work well for someone versed in the theories and properties, but most people taking the GMAT (and those that read these forums that do not ever post) are not ones that know or even care to know the ins and outs of deep theories. The number picking on the GMAT works well, is easy, and you just have to understand what the MUST BE TRUE means, that if there is a situation where the option is greater than 2, that does not mean that it always will be. You have to consider numerous items. All I"m saying is that for the majority of people, spending 2 min plugging in numbers will keep them better focused, working towards a solution, and on track for the rest of the GMAT. That's the most important part. _________________ ------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

GMAT Club Premium Membership - big benefits and savings

CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2795
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35
Followers: 229

Kudos [?]: 1646 [1] , given: 235

### Show Tags

13 Oct 2009, 13:53
1
KUDOS
In exams they donot expect us to spend much time and this is a tricky question.

DONOT use number method as your first attempt.

Solution:

1/sqrt(x+y) = sqrt(x+y)/x+y // multiply den and numerator by sqrt(x+y)

For 1st option we cannot say anything.
Now for 2nd and third denominators are same that means we need to consider only numerator,and rem they have stated A n B both r positive.

now its obvious that sqrt x + sqrt y > sqrt (x+y) > sqrt x - sqrt y
( use the property.... (a+b)^2 = a^2 + b^2 + 2ab )

If any doubts pls letme know.This question shouldnt take more than 1 min if you just concentrate on basic knwledge
_________________

Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html

SVP
Joined: 17 Feb 2010
Posts: 1558
Followers: 19

Kudos [?]: 597 [0], given: 6

### Show Tags

04 Aug 2010, 12:04
Hey Bunuel,

For this problem, if we were to test by picking numbers then ideally we should check with integers as well as with fractions...right?? Because the problem does not say that we have to stick with integers only.
Senior Manager
Status: Time to step up the tempo
Joined: 24 Jun 2010
Posts: 408
Location: Milky way
Schools: ISB, Tepper - CMU, Chicago Booth, LSB
Followers: 8

Kudos [?]: 199 [3] , given: 50

Re: Must be greater than.. [#permalink]

### Show Tags

10 Oct 2010, 15:33
3
KUDOS
TehJay wrote:
If x and y are positive, which of the following must be greater than $$\frac{1}{\sqrt{x+y}}$$?

I. $$\frac{\sqrt{x+y}}{2x}$$

II. $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$

III. $$\frac{\sqrt{x}-\sqrt{y}}{x+y}$$

(A) None
(B) I only
(C) II only
(D) I and III
(E) II and III

Since this question is a must be true type. If we can find even one scenario wherein the condition does not hold for either I, II or III we can eliminate that choice.

Picking numbers as x=1 and y=1, we can see that only the II option satisfies the condition.

$$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$ > $$\frac{1}{\sqrt{x+y}}$$
Since $$1>\frac{1}{\sqrt{2}}$$. Answer is C.
_________________

Support GMAT Club by putting a GMAT Club badge on your blog

Ms. Big Fat Panda
Status: Three Down.
Joined: 09 Jun 2010
Posts: 1922
Concentration: General Management, Nonprofit
Followers: 452

Kudos [?]: 2003 [0], given: 210

Re: PS: If x and y are positive, [#permalink]

### Show Tags

23 Nov 2010, 22:27
vrajesh wrote:
If x and y are positive, which of the following must be greater than $$\frac{1}{\sqrt{x+y}}?$$

I. $$\frac{\sqrt{x+y}}{2x}$$

II.$$\frac{\sqrt{x} + \sqrt{y}}{\sqrt{x+y}}$$

III. $$\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x+y}}$$

A. None
B. I only
C. II only
D. I and III
E. II and III

Can someone please explain what is the best way to solve this problem in under 2 minutes?

I would really like to understand how to solve these type of problems.

What makes this problem especially difficult, is the condition x and y are positive
i.e. x > 0 and y > 0

hence x and y can be any of the following 1/2, 3/4, 1, 2, 3, 4, ....., and more

Okay, so the main fraction can be simplified to the following by means of rationalization.

$$\frac{1}{\sqrt{x+y}}$$ = $$\frac{\sqrt{x+y}}{x+y}$$

Now the question asks for something that MUST be true for all the values of x and y, which are positive.

So look at the options:

I: Here, your denominator is 2x. If x+y > 2x, then this option is greater than the given number. If not, it's smaller. So this can't be the answer.

II: This always has to be greater. In our simplified form, our numerator is 1, and here the numerator is $$\sqrt{x} + \sqrt{y}$$ which has to be greater than 1, since the smallest possible value that x can take is 1 (Remember 0 is not a positive integer) - So this option is good.

III. $$\sqrt{x} - \sqrt{y}$$. This can be greater than or lesser than one depending on the values that x and y takes, so this need not ALWAYS be greater than what's given to us. Hence incorrect.

Thus the final answer is C, just option II. Hope this helps.
Intern
Joined: 25 Jul 2009
Posts: 11
Followers: 0

Kudos [?]: 3 [0], given: 2

Re: PS: If x and y are positive, [#permalink]

### Show Tags

23 Nov 2010, 23:00
How do you handle for fractions, since fractions are still positive??
_________________

Failure it not and option -- Gene Kranz

Ms. Big Fat Panda
Status: Three Down.
Joined: 09 Jun 2010
Posts: 1922
Concentration: General Management, Nonprofit
Followers: 452

Kudos [?]: 2003 [0], given: 210

Re: PS: If x and y are positive, [#permalink]

### Show Tags

23 Nov 2010, 23:05
What do you mean for fractions?

The question asks about the options that MUST be greater. So even if you can find one positive x or y for which the fraction in the answer is lesser than the fraction given, your answer choice is ruled out.
Intern
Joined: 25 Jul 2009
Posts: 11
Followers: 0

Kudos [?]: 3 [0], given: 2

Re: PS: If x and y are positive, [#permalink]

### Show Tags

23 Nov 2010, 23:09
Since x is positive x can be 1/2 and same goes for y. Y can be 2/3 or some other fraction since the questionb does say x and y are integers
_________________

Failure it not and option -- Gene Kranz

Ms. Big Fat Panda
Status: Three Down.
Joined: 09 Jun 2010
Posts: 1922
Concentration: General Management, Nonprofit
Followers: 452

Kudos [?]: 2003 [0], given: 210

Re: PS: If x and y are positive, [#permalink]

### Show Tags

23 Nov 2010, 23:17
Ah, good point. Are you sure that the question doesn't say it's an integer? I didn't consider that possibility, for option II. There must be some way to prove that it works even for fractions.
SVP
Joined: 16 Nov 2010
Posts: 1672
Location: United States (IN)
Concentration: Strategy, Technology
Followers: 33

Kudos [?]: 522 [0], given: 36

### Show Tags

30 Apr 2011, 19:53
Let x = 2, y = 2

1/sqrt(2+2) = 1/sqrt(4) = 1/2

(sqrt(x+y))/2x = sqrt(4)/4 = 1/2

(sqrtx + sqrty)/(x+y) = 2sqrt(2)/4 = sqrt(2)/2 = 1/sqrt(2) > 1/2

sqrt(2) - sqrt(2)/(2+2) = 0

_________________

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

GMAT Club Premium Membership - big benefits and savings

Intern
Joined: 16 Jan 2012
Posts: 4
Followers: 0

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

Re: If x and y are positive, which of the following must be [#permalink]

### Show Tags

14 Mar 2012, 08:07
Picking numbers actually got me the wrong answer
x= 9
y=16

1/sqrt(x+y)== 1/5===0.20
1) sqrt(x+y)/2x = 5/18= 0.2777

Math Expert
Joined: 02 Sep 2009
Posts: 37144
Followers: 7273

Kudos [?]: 96806 [0], given: 10786

Re: If x and y are positive, which of the following must be [#permalink]

### Show Tags

14 Mar 2012, 08:18
Ashamock wrote:
Picking numbers actually got me the wrong answer
x= 9
y=16

1/sqrt(x+y)== 1/5===0.20
1) sqrt(x+y)/2x = 5/18= 0.2777

The question asks which of the options MUST be greater than $$\frac{1}{\sqrt{x+y}}$$, not COULD be greater than $$\frac{1}{\sqrt{x+y}}$$. Hence one set of numbers showing that option (1) is greater is not enough to conclude that this option is ALWAYS greater (greater for all numbers).

Hope it's clear.
_________________
Intern
Joined: 26 Nov 2011
Posts: 15
Followers: 0

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

Re: If x and y are positive, which of the following must be [#permalink]

### Show Tags

02 Jul 2012, 03:09
Hi,

Please correct me where I am wrong.

I understood why I and III are false.

If {x,y} = {2,2} then 1/(x+y)^1/2 = 1/2 = 0.5

while II will come out to be 1/2*2^1/2 = 0.357

Here II is not greater than the given expression.

So, None should be an answer. isnt it?
Math Expert
Joined: 02 Sep 2009
Posts: 37144
Followers: 7273

Kudos [?]: 96806 [0], given: 10786

If x and y are positive, which of the following must be [#permalink]

### Show Tags

02 Jul 2012, 04:59
nishantmehra01 wrote:
Hi,

Please correct me where I am wrong.

I understood why I and III are false.

If {x,y} = {2,2} then 1/(x+y)^1/2 = 1/2 = 0.5

while II will come out to be 1/2*2^1/2 = 0.357

Here II is not greater than the given expression.

So, None should be an answer. isnt
it?

If $$x=y=2$$, then:

$$\frac{1}{\sqrt{x+y}}=\frac{1}{2}$$ and $$\frac{\sqrt{x}+\sqrt{y}}{x+y}=\frac{\sqrt{2}+\sqrt{2}}{2+2}=\frac{1}{\sqrt{2}}$$ --> $$\frac{1}{2}<\frac{1}{\sqrt{2}}$$.

Hope it's clear.
_________________
Intern
Joined: 08 Mar 2013
Posts: 19
Followers: 0

Kudos [?]: 4 [0], given: 7

### Show Tags

03 May 2013, 19:22
Bunuel wrote:

SOLUTION #2
The method called cross multiplication:
Suppose we want to know which [b]positive
fraction is greater $$\frac{9}{11}$$ or $$\frac{13}{15}$$: crossmultiply $$9*15=135$$ and $$11*13=143$$ --> $$135<143$$ which fraction gave us numerator for bigger value 143? $$\frac{13}{15}$$! Thus $$\frac{13}{15}>\frac{9}{11}$$.

Lets do the same with our problem:
I. $$\frac{\sqrt{x+y}}{2x}$$ and $$\frac{1}{\sqrt{x+y}}$$ --> $$\sqrt{x+y}*\sqrt{x+y}=x+y$$ and $$2x*1=2x$$. $$x+y$$ may or may not be greater than $$2x$$. OUT

II. $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$ and $$\frac{1}{\sqrt{x+y}}$$ --> $$(\sqrt{x}+\sqrt{y})(\sqrt{x+y})$$ and $$x+y$$. Divide both sides by $$\sqrt{x+y}$$ --> $$\sqrt{x}+\sqrt{y}$$ and $$\sqrt{x+y}$$. We know that $$\sqrt{x}+\sqrt{y}$$ is always greater, which one gave the numerator for it: $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$, so $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$ is always greater than $$\frac{1}{\sqrt{x+y}}$$. OK

III. Well we can not even consider this one as our expression $$\frac{1}{\sqrt{x+y}}$$ is always positive and the $$\sqrt{x}-\sqrt{y}$$ (numerator) can be negative. OUT

Hope it's clear.

Major thanks for this one! I'm really bad with these types of problems, and hate plugging in numbers, but I'm very good with algebraic equations, and the cross-multiplication method just naturally makes a lot of sense to me. I think if I do a few dozen of these, there's no chance I'd go wrong on these types of questions on GMAT!
Manager
Joined: 03 Mar 2013
Posts: 91
Location: India
Concentration: General Management, Marketing
GPA: 3.49
WE: Web Development (Computer Software)
Followers: 0

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

Re: If x and y are positive, which of the following must be [#permalink]

### Show Tags

04 Jul 2013, 18:43
study wrote:
If x and y are positive, which of the following must be greater than $$\frac{1}{\sqrt{x+y}}$$?

1. $$\frac{\sqrt{x+y}}{2x}$$

2. $$\frac{\sqrt{x}+\sqrt{y}}{x+y}$$

3. $$\frac{\sqrt{x}-\sqrt{y}}{x+y}$$

(A) None
(B) 1 only
(C) 2 only
(D) 1 and 3 only
(E) 2 and 3 only

plugin values

put x = 3 and y = 1, u may also try out x = 2 and y = 2 but we get zero in one choice, which may again hold one's horses with doubt,

once u substitute we get only c as our answer and lastly remember we are looking for choices which are greater than but not equal to
Intern
Joined: 04 May 2013
Posts: 47
Followers: 0

Kudos [?]: 7 [0], given: 7

Re: If x and y are positive, which of the following must be [#permalink]

### Show Tags

08 Jul 2013, 14:57
I mess up while using numbers in square roots and for me it takes longer usually.

I just did algebra which was quicker for me.
Use conjugation
I think Bunuel did this is his post but not sure:

1/√(x+y) * (1/1)
= [1/√(x+y)] * [√(x+y)/√(x+y)]
= √(x+y)/(x+y) (Because multiplying the square roots gets rid of these roots)

From this, we can clearly see which one is greater than the equation mentioned.

For 2, since denominator are the same, we only need to look at the numerator, and

√(x) + √(y) > √(x+y) -- Always.

The other are clearly smaller for the same reason.
Re: If x and y are positive, which of the following must be   [#permalink] 08 Jul 2013, 14:57

Go to page    1   2    Next  [ 35 posts ]

Similar topics Replies Last post
Similar
Topics:
If x is positive and y is negative, which of the following must be 1 11 Apr 2016, 03:03
2 If x is negative and y is positive, which of the following must be 7 08 Nov 2015, 23:31
If x is negative, which of the following must be positive ? 1 08 Nov 2015, 23:11
15 If x and y are positive odd integers, then which of the following must 8 26 Feb 2015, 05:53
85 If x and y are positive, which of the following must be 46 08 Aug 2009, 06:57
Display posts from previous: Sort by