It is currently 23 Nov 2017, 20:33

Close

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
Your Progress

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.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Director
Director
User avatar
Joined: 07 Aug 2011
Posts: 579

Kudos [?]: 548 [0], given: 75

Concentration: International Business, Technology
GMAT 1: 630 Q49 V27
GMAT ToolKit User
Re: If x and y are positive, which of the following must be [#permalink]

Show Tags

New post 17 Jan 2015, 08:49
Bunuel wrote:
mmcooley33 wrote:
walker wrote:
Let's consider original statement: \(\frac{1}{\sqrt{x+y}}\)

How can we approach the problem fast? Let's see when the original statement is very large : x,y ---> 0 and \(\frac{1}{\sqrt{x+y}}\) goes to infinity.
Now, let's see what do our options at x,y ---> 0.

I) \(\frac{\sqrt{x+y}}{2}\) goes to 0 at x,y ---> 0.

II) \(\frac{\sqrt{x}+\sqrt{y}}{2}\) goes to 0 at x,y ---> 0.

III) \(\frac{\sqrt{x}-sqrt{y}}{x+y}\) hm... it has x+y as a denominator. But what if x=y? At x=y it equals 0.

So, none of the options.


Can you explain how you used this in more detail, and perhaps give me some idea of how to apply this same principle going forward. Thanks a lot, i appreciate the help.


Just to add couple of words to Walker's great solution:

Note that we are asked "which of the following MUST be greater than \(\frac{1}{\sqrt{x+y}}\)?" not COULD be greater.

"MUST BE TRUE" questions:
These questions ask which of the following MUST be true (must be greater in our case), or which of the following is ALWAYS true no matter what set of numbers you choose. Generally for such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

As for "COULD BE TRUE" questions:
The questions asking which of the following COULD be true are different: if you can prove that a statement is true for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.

So, if we find even one set of \(x\) and \(y\) for which \(\frac{1}{\sqrt{x+y}}\) is greater than option I for example then it'll mean that option I is not ALWAYS greater then \(\frac{1}{\sqrt{x+y}}\).

How can we increase the value of \(\frac{1}{\sqrt{x+y}}\)? Testing extreme examples: if \(x\) and \(y\) are very small (when their values approach zero) then denominator approaches zero and thus the value of the fraction goes to +infinity, becomes very large. In this case:

I. \(\frac{\sqrt{x+y}}{2}\): nominator approaches zero, so the value of the whole fraction approaches zero as well, becoming very small. So this option is not always more than given fraction;
II. \(\frac{\sqrt{x}+\sqrt{y}}{2}\): the same here - nominator approaches zero, so the value of the whole fraction approaches zero as well, becoming very small. So this option is not always more than given fraction;

As for:
III. \(\frac{\sqrt{x}-\sqrt{y}}{x+y}\), if \(x=y\) then this fraction equals to zero and \(\frac{1}{\sqrt{x+y}}\) has some value more than zero, so this option also is not always more than given fraction;

Answer: none of the options must be greater than the given fraction.

Hope it's clear.


Thanks Bunuel walker . Thanks for pointing out use of extreme cases for ruling out options.
_________________

Thanks,
Lucky

_______________________________________________________
Kindly press the Image to appreciate my post !! :-)

Kudos [?]: 548 [0], given: 75

Intern
Intern
avatar
Joined: 26 Mar 2013
Posts: 21

Kudos [?]: 15 [0], given: 2

Location: India
Concentration: Finance, Strategy
Schools: Booth PT '18 (S)
Re: If x and y are positive, which of the following must be [#permalink]

Show Tags

New post 17 Jan 2015, 12:26
Ans E

stat 1 & 2 were straight....but 3 though was easily simplified....took smtym to confirm

Good tricky qtn

Kudos [?]: 15 [0], given: 2

Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 15506

Kudos [?]: 283 [0], given: 0

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

Show Tags

New post 27 Jan 2016, 13:16
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.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Kudos [?]: 283 [0], given: 0

Current Student
avatar
B
Joined: 20 Mar 2014
Posts: 2676

Kudos [?]: 1775 [0], given: 794

Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
GMAT ToolKit User Premium Member Reviews Badge
If x and y are positive, which of the following must be [#permalink]

Show Tags

New post 28 Jan 2016, 10:41
noboru wrote:
If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?

I. \(\frac{\sqrt{x+y}}{2}\)
II. \(\frac{\sqrt{x}+\sqrt{y}}{2}\)
III. \(\frac{\sqrt{x}-\sqrt{y}}{x+y}\)

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


In "MUST BE TRUE" questions, you need to use POE and be ready to prove that none of the options/statements are possible. Even 1 not possible scenario will make that option not allowed.

Had this question been "could be true" instead, II only would have been correct with x=y=1. But as this is a MUST BE TRUE question, you need to make sure to find 1 set of (x,y) that will negate the given conditions.

i) and iii) can be easily POE-d by assuming x=y=1. In both these cases the resulting values will NOT BE GREATER than \(1/(x+y)^{0.5}\).

For ii), you can clearly see x=y=1 gives you a value greater than \(1/(x+y)^{0.5}\) but what about x=y=4 again you get a value greater. So lets take x=y=0.25. In this case you will end up getting ii) < \(1/(x+y)^{0.5}\). Hence this expression as well is NOT always true and is hence eliminated.

As you eliminated all the 3 possible options, OA must be E (none).

Hope this helps.

Kudos [?]: 1775 [0], given: 794

Intern
Intern
avatar
Joined: 30 Sep 2016
Posts: 8

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

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

Show Tags

New post 29 Nov 2016, 07:23
Hi.

Would you mind explaining why I got the wrong answer? I assumed x as 3 and y as 6 as we know they are positive integers. Is it wrong for me to assume this?

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

Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42338

Kudos [?]: 133160 [0], given: 12415

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

Show Tags

New post 29 Nov 2016, 07:43
Nasahtahir wrote:
Hi.

Would you mind explaining why I got the wrong answer? I assumed x as 3 and y as 6 as we know they are positive integers. Is it wrong for me to assume this?


The question asks which of the following MUST be greater than ... So, which is ALWAYS greater than ... If it's greater for some particular set of numbers it does not mean that it will be greater for other sets.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 133160 [0], given: 12415

Board of Directors
User avatar
G
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 3102

Kudos [?]: 1119 [0], given: 327

Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
GMAT ToolKit User Premium Member
Re: If x and y are positive, which of the following must be [#permalink]

Show Tags

New post 29 Nov 2016, 09:13
noboru wrote:
If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?

I. \(\frac{\sqrt{x+y}}{2}\)
II. \(\frac{\sqrt{x}+\sqrt{y}}{2}\)
III. \(\frac{\sqrt{x}-\sqrt{y}}{x+y}\)

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


Plug in an try -

\(x = 9\) & \(y = 16\)

\(\frac{1}{\sqrt{x+y}}\) = \(\frac{1}{25}\) \(= 0.04\)


I. \(\frac{\sqrt{x+y}}{2}\) \(= \frac{5}{2} =2.5\)

II. \(\frac{\sqrt{x}+\sqrt{y}}{2}\) \(= \frac{3 + 4}{2}\) \(= 3.5\)

Option III is a bit different , there can be 2 cases -

\(x = 9\) & \(y = 16\) & \(x = 16\) & \(y = 9\)

III. \(\frac{\sqrt{x}-\sqrt{y}}{x+y}\)

Thus there can be multiple possible solutions for option (III)

Hence, we are confident about I & II, but option III , may or may not be > \(\frac{1}{\sqrt{x+y}}\) , so answer will be (E) None of the above.
_________________

Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )

Kudos [?]: 1119 [0], given: 327

Expert Post
Top Contributor
SVP
SVP
User avatar
G
Joined: 12 Sep 2015
Posts: 1851

Kudos [?]: 2627 [0], given: 362

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

Show Tags

New post 25 Oct 2017, 07:13
Expert's post
Top Contributor
noboru wrote:
If x and y are positive, which of the following must be greater than \(\frac{1}{\sqrt{x+y}}\)?

I. \(\frac{\sqrt{x+y}}{2}\)
II. \(\frac{\sqrt{x}+\sqrt{y}}{2}\)
III. \(\frac{\sqrt{x}-\sqrt{y}}{x+y}\)

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


Let's test some values.

x = 1 and y = 1
1/√(x + y) = 1/√(1 + 1) = 1/√2

I. √(x + y)/2 = √(1 + 1)/2 = √2/2
Notice that, if we take 1/√2 and multiply top and bottom by √2, we get: √2/2, which is the same as quantity I
Since quantity I is not greater than 1/√2, statement I is not true

II. (√x + √y)/2 = (√1 + √1)/2 = (1 + 1)/2 = 2/2 = 1
Since 1 IS greater than 1/√2, we cannot say for certain whether quantity II will always be greater than √(x + y)/2

III. (√x - √y)/(x + y) = (√1 - √1)/(1 + 1) = (1 - 1)/2 = 0/2 = 0
Since 0 is not greater than 1/√2, statement III is not true

So, statements I and III are definitely not true, and we aren't yet 100% certain about statement II
Let's try another pair of values for x and y


x = 0.25 and y = 0.25
1/√(x + y) = 1/√(0.25 + 0.25) = 1/√0.5
Let's further simplify 1/√0.5
Since 1 = √1, we can say: √1/√0.5
Then we'll use a rule that says (√k)/(√j) = √(k/j)
So, √1/√0.5 = √(1/0.5) = √2
We see that, when x = 0.25 and y = 0.25, 1/√(x + y) = √2

II. (√x + √y)/2 = (√0.25 + √0.25)/2 = (0.5 + 0.5)/2 = 1/2
Since 1/2 is NOT greater than √2, statement II is not true

Answer:
[Reveal] Spoiler:
E

_________________

Brent Hanneson – Founder of gmatprepnow.com

Image

Kudos [?]: 2627 [0], given: 362

Re: If x and y are positive, which of the following must be   [#permalink] 25 Oct 2017, 07:13

Go to page   Previous    1   2   3   [ 48 posts ] 

Display posts from previous: Sort by

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.