Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

It is currently 21 Jul 2019, 05:59

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

If x and y are positive integers such that x < y, which of the followi

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

Hide Tags

Find Similar Topics 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56306
If x and y are positive integers such that x < y, which of the followi  [#permalink]

Show Tags

New post 29 Mar 2018, 00:21
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

65% (01:42) correct 35% (01:17) wrong based on 111 sessions

HideShow timer Statistics


examPAL Representative
User avatar
P
Joined: 07 Dec 2017
Posts: 1073
Re: If x and y are positive integers such that x < y, which of the followi  [#permalink]

Show Tags

New post Updated on: 29 Mar 2018, 10:03
Bunuel wrote:
If x and y are positive integers such that x < y, which of the following expressions must be less than 1?


I. \(\sqrt{\frac{y}{x}}\)

II. \(\frac{x^2 - 100}{y^2 - 100}\)

III. \(\frac{x}{y}\)


(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only


As there are specific rules that govern if a fraction is larger or smaller than 1, we'll use them.
This is a Precise approach.

A fraction is smaller than 1 if its numerator is smaller than its denominator.
That is, \(\frac{smaller}{larger}<1\). Since x<y and both are positive, then \(\frac{x}{y}<1\).
III is true so (A), (B), (D) are eliminated.
So all we need to know is if II is true. Since x < y and both are positive integers then x^2 < y^2 meaning that x^2 - 100 < y^2 - 100.
So, if both are positive then \(\frac{x^2 - 100}{y^2 - 100}\) is smaller than 1.
But what happens if both numerator and denominator are negative? In this case, then the numberator is a 'smaller number' = 'larger negative' and the denominator a 'larger number' = 'smaller negative' and once we cancel out the minus sign we get a fraction whose numerator is larger than its denominator, meaning that it is larger than 1.

(C) s our answer.
_________________

Originally posted by DavidTutorexamPAL on 29 Mar 2018, 00:49.
Last edited by DavidTutorexamPAL on 29 Mar 2018, 10:03, edited 1 time in total.
Intern
Intern
avatar
B
Joined: 02 Jul 2017
Posts: 2
Re: If x and y are positive integers such that x < y, which of the followi  [#permalink]

Show Tags

New post 29 Mar 2018, 09:56
DavidTutorexamPAL wrote:
Bunuel wrote:
If x and y are positive integers such that x < y, which of the following expressions must be less than 1?


I. \(\sqrt{\frac{y}{x}}\)

II. \(\frac{x^2 - 100}{y^2 - 100}\)

III. \(\frac{x}{y}\)


(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only


As there are specific rules that govern if a fraction is larger or smaller than 1, we'll use them.
This is a Precise approach.

A fraction is smaller than 1 if its numerator is smaller than its denominator.
That is, \(\frac{smaller}{larger}<1\). Since x<y and both are positive, then \(\frac{x}{y}<1\).
III is true so (A), (B), (D) are eliminated.
So all we need to know is if II is true. Since x < y and both are positive integers then x^2 < y^2 meaning that x^2 - 100 < y^2 - 100.
So, if both are positive then \(\frac{x^2 - 100}{y^2 - 100}\) is smaller than 1.
But what happens if both are negative? In this case, then the numberator is a 'smaller number' = 'larger negative' and the denominator a 'larger number' = 'smaller negative' and once we cancel out the minus sign we get a fraction whose numerator is larger than its denominator, meaning that it is larger than 1.

(C) s our answer.


The question states that x and y are positive integers, hence our answer will be (E)
examPAL Representative
User avatar
P
Joined: 07 Dec 2017
Posts: 1073
Re: If x and y are positive integers such that x < y, which of the followi  [#permalink]

Show Tags

New post 29 Mar 2018, 10:01
qwertybd wrote:
The question states that x and y are positive integers, hence our answer will be (E)


Try x = 2 and y = 3.
Then:
x^2 - 100 = -96
y^2 - 100 = -91.

Their quotient is 96/91 which is larger than 1.

I think the problem came from lack of clarity in my answer: 'both are negative' doesn't refer to x and y but to the expressions (x^2 - 100) and (y^2 - 100)
I edited the original answer for clarity, thanks!
_________________
Intern
Intern
User avatar
B
Joined: 05 Dec 2017
Posts: 20
GPA: 3.35
Re: If x and y are positive integers such that x < y, which of the followi  [#permalink]

Show Tags

New post 10 Apr 2018, 22:50
1
Hi, Bunuel sir. we expect your clarification for answer of the above question.
Why the answer shows the option E.
Where, it is easily found that x=2, y=3, so (x^2 - 100)/(y^2 -100) >1.
So ans should be C only
Then based on which cogent logic it should be the answer option E.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56306
Re: If x and y are positive integers such that x < y, which of the followi  [#permalink]

Show Tags

New post 10 Apr 2018, 23:41
GMAT Club Bot
Re: If x and y are positive integers such that x < y, which of the followi   [#permalink] 10 Apr 2018, 23:41
Display posts from previous: Sort by

If x and y are positive integers such that x < y, which of the followi

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





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