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

 It is currently 19 Aug 2018, 00:38

### 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

# Let ] represent the average of the greatest integer less

Author Message
TAGS:

### Hide Tags

Senior Manager
Status: 750+ or Burst !
Joined: 01 May 2011
Posts: 367
Location: India
Concentration: General Management, Strategy
GMAT 1: 670 Q48 V35
GPA: 3.5
Let ] represent the average of the greatest integer less  [#permalink]

### Show Tags

17 Sep 2011, 01:07
2
2
00:00

Difficulty:

75% (hard)

Question Stats:

54% (02:12) correct 46% (01:41) wrong based on 105 sessions

### HideShow timer Statistics

Let [[x]] represent the average of the greatest integer less than or equal to x and the least integer greater than or equal to x . Is $$0=< |a| =<1$$?

(1) [[x]] - x = a
(2) $$0< [[a]]<1$$?

_________________

GMAT done - a mediocre score but I still have a lot of grit in me

The last 20 days of my GMAT journey

Intern
Joined: 17 Aug 2011
Posts: 5

### Show Tags

17 Sep 2011, 02:05
akhileshgupta05 wrote:
Let [[x]] represent the average of the greatest integer less than or equal to x and the least integer greater than or equal to x . Is $$0=< |a| =<1$$?

(1) [[x]] - x = a
(2) $$0< [[a]]<1$$?

x - k1 : the greatest integer less than or equal to x
x + k2 : the least integer greater than or equal to x
--> 0=< k1,k2 <=1 and [[x]] = (x-k1+x+k2)/2 = (2x+ k2-k1)/2 = x + (k2-k1)/2

(1) [[x]] - x = (k2-k1)/2 Or a=(k2-k1)/2
Because 0=< k1,k2 <=1 --> 0=< | (k2-k1)/2 | <=1 or 0=<|a|<=1
Hence, (1) is suff.

(2) Similar to [[x]] : [[a]] = a+ (r2-r1)/2 in which 0=<r1,r2<=1
0<[[a]]<1
0< (a-r1+a+r2)/2 <1
0< a-r1+a+r2 <2
Because a-r1 and a+ r2 are interger, a-r1+a+r2 =1
--> 2a = 1- (r2-r1)
--> a = 1/2-(r2-r1)/2
0=<r1,r2<=1 Hence, 0=<|a|<=1
(2) is suff

Manager
Joined: 31 May 2011
Posts: 81
Location: India
GMAT Date: 12-07-2011
GPA: 3.22
WE: Information Technology (Computer Software)

### Show Tags

17 Sep 2011, 02:35
1
akhileshgupta05 wrote:
Let [[x]] represent the average of the greatest integer less than or equal to x and the least integer greater than or equal to x . Is $$0=< |a| =<1$$?

(1) [[x]] - x = a
(2) $$0< [[a]]<1$$?

The answer should be D. This is because:

for any value of x, the difference between the greatest integer less than or equal to x AND the least integer greater than or equal to x will always vary between 1 unit. hence
1)
=> -1<= [[x]] - x <=1
or -1<= a <= 1
or 0 <= |a| <=1

hence sufficient

2) again as mentioned above and given:0< [[a]] < 1.

This will be possible only when x is between 0 and 1 so that we get he greatest value as one and least value as 0. This will only result in 0< [[a]] < 1, as [[a]] is av. of greatest and the least integer.
hence:
0 <= a <=1 and hence 0 <= |a| <=1.
sufficient
CEO
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2653
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35

### Show Tags

19 Sep 2011, 09:00
For such daunting questions, instead of looking for a solution visually better write down what is given and what is required.

Suppose the number x = n.m ( if n.m = 2.1 => n = 2 and m =1)

[[x]] = (n+n+1)/2 = n +1/2 because greatest integer is n+1 and smallest is n

1. [[x]] - x = a
=> n+1/2 -n.m = a
=> a = n+1/2 - n - 0.m = 1/2 - 0.m
The absolute value of a will always oscillate between 0 and 1. Hence sufficient.

2. 0< [[a]] <1 => 0 < a+1/2 < 1
=> -1/2 < a < 1/2
=> the value of a oscillates between -1/2 and 1/2 => the absolute value lies between 0 and 1/2.
Hence sufficient.

Hence D
_________________

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

Intern
Status: Target 660 -> 720(Q49,V41) retaking in Feb 2012
Joined: 22 Aug 2011
Posts: 6
Location: United States
GMAT 1: 660 Q47 V34
WE: Information Technology (Computer Software)

### Show Tags

20 Sep 2011, 08:38
1)Simpliy the statement
Greatest int <=x is x
Least int >=x is x
Hence [[x]] = x
2)Evaluate 1.
[[x]] - x = a
x-x = 0 so a = 0
1. is Sufficient
4)Evaluate 2.
0<[[a]]<1
0<a<1
2. is sufficient

Ans D
Retired Moderator
Joined: 20 Dec 2010
Posts: 1877

### Show Tags

20 Sep 2011, 11:37
parulbhatnagar wrote:
it's not so difficult question
i think [[x]] is nothing but x itself, so dont get confused

My answer - D (both suff.)

x=2.3456; [[x]]=2.5
x=0; [[x]=0
x=-2.3456; [[x]]=-2.5

So, your statement is true only if x=integer.
_________________
VP
Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 1124

### Show Tags

21 Sep 2011, 22:54
check for x= 0.4 | 1.6 |-0.4 | -1.7

a for 0.4, 0.5-0.4=0.1 Y ; for 1.6, 1.5-1.6= -0.1 Y
for -0.4, -0.5+0.4 = 0.1 Y ; for -1.7, -1.5+1.7 = 0.2 Y

thus sufficient.

b satisfies for 0 <= a <= 1 hence sufficient.

D it is.
_________________

Visit -- http://www.sustainable-sphere.com/
Promote Green Business,Sustainable Living and Green Earth !!

Senior Manager
Status: MBAing!!!!
Joined: 24 Jun 2011
Posts: 252
Location: United States (FL)
Concentration: Finance, Real Estate
GPA: 3.65
WE: Project Management (Real Estate)

### Show Tags

29 Sep 2011, 13:03
It was hard to understand the question. I picked A initially but then I agreed that D was the answer.

Pick some numbers first and draw a numbers line. x=2.3 then the lower integer= 2, upper integer=3 therefore [[2.3]]=2.5

stm 1: [[2.3]]-2.3=a
2.5-2.3=a
0.2=a....sufficient

stm 2: if [[a]]=average of the upper and lower integers and it is between 0 and 1 then a must be between 0 and 1....sufficient
Manager
Affiliations: Project Management Professional (PMP)
Joined: 30 Jun 2011
Posts: 166
Location: New Delhi, India
Re: Let [[x]] represent the average of the greatest integer less  [#permalink]

### Show Tags

18 May 2012, 06:45
3
Smita04 wrote:
Let [[x]] represent the average of the greatest integer less than or equal to x and the least integer greater than or equal to x. Is 0 <= |a| <= 1 ?
Hi Smita

1) [[x]] - x = a
2) 0 < [[a]] < 1

This is a good one.

Let's try to understand the meaning of [[x]].
If x = 4.1, [[x]] = (4 + 5)/2 = 4.5
If x = 3.9, [[x]] = (3 + 4)/2 = 3.5

So, if x = Integer + d (where 'I' is its decimal part whereas 'd' is its decimal part),
then [[x]] = I + 0.5

Statement(1):
[[x]] - x = (I + 0.5) - (I + d) = 0.5 - d
|0.5 - d| will always be between 0 and 1, since d is between 0 and 0.999...
SUFFICIENT.

Statement(2):
0< [[a]] < 1
0 < I + d < 1
Since the decimal value is always between 0 and 1, the integral value of a = 0
So,
0 < a < 1
or
0 <= |a| <= 1
SUFFICIENT.
_________________

Best
Vaibhav

If you found my contribution helpful, please click the +1 Kudos button on the left, Thanks

Intern
Joined: 13 Mar 2012
Posts: 15
GMAT 1: 700 Q50 V34
Re: Let [[x]] represent the average of the greatest integer less  [#permalink]

### Show Tags

19 May 2012, 09:45
1
Smita04 wrote:
Let [[x]] represent the average of the greatest integer less than or equal to x and the least integer greater than or equal to x. Is 0 <= |a| <= 1 ?

(1) [[x]] - x = a
(2) 0 < [[a]] < 1

consider 1: 0<a<1/2
consider 2: --> a=1/2,
in both cases , sufficient,
thus D
Intern
Joined: 27 Oct 2011
Posts: 13
Schools: Cambridge
Re: Let [[x]] represent the average of the greatest integer less  [#permalink]

### Show Tags

22 May 2012, 10:55
[[x]] = x when x is integer.==> a=0

[[x]]= [x]+.5 when x is a not integer.==>a=[x]+.5 -x = -{x}+.5 which always lies in between -.5 to .5
so |a|<1.
Director
Joined: 22 Mar 2011
Posts: 604
WE: Science (Education)
Re: Let [[x]] represent the average of the greatest integer less  [#permalink]

### Show Tags

Updated on: 04 Jul 2012, 11:55
riteshgupta wrote:
Bunuel can you please explain this question, esp. the second part. Thanks in advance

I can try to explain it:
Any real number is situated between two consecutive integers. So, there is an integer k such that $$k \leq a < k+1$$.
Then [[a]] = k+0.5, and combining with (2), we get that 0 < k+0.5 < 1, which means -0.5 < k < 0.5. Being an integer,
it follows that k must be 0. Therefore, $$0 \leq a < 1$$, and (2) is sufficient.
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Originally posted by EvaJager on 04 Jul 2012, 10:28.
Last edited by EvaJager on 04 Jul 2012, 11:55, edited 1 time in total.
Director
Joined: 22 Mar 2011
Posts: 604
WE: Science (Education)
Re: Let [[x]] represent the average of the greatest integer less  [#permalink]

### Show Tags

04 Jul 2012, 11:57
riteshgupta wrote:
EvaJager wrote:
riteshgupta wrote:
Bunuel can you please explain this question, esp. the second part. Thanks in advance

I can try to explain it:
Any real number is situated between two consecutive integers. So, there is an integer k such that $$k \leq a \leq k+1$$.
Then [[a]] = k+0.5, and combining with (2), we get that 0 < k+0.5 < 1, which means -0.5 < k < 0.5. Being an integer,
it follows that k must be 0. Therefore, $$0 \leq a \leq 1$$, and (2) is sufficient.

Thanks EvaJager...

Very welcome.

Just slight corrections: it should be $$k \leq a < k+1$$ and $$0 \leq a < 1$$.
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Intern
Joined: 14 Jan 2013
Posts: 3
Re: Let [[x]] represent the average of the greatest integer less  [#permalink]

### Show Tags

02 Aug 2013, 13:52
narangvaibhav wrote:
Smita04 wrote:
Let [[x]] represent the average of the greatest integer less than or equal to x and the least integer greater than or equal to x. Is 0 <= |a| <= 1 ?
Hi Smita

1) [[x]] - x = a
2) 0 < [[a]] < 1

This is a good one.

Let's try to understand the meaning of [[x]].
If x = 4.1, [[x]] = (4 + 5)/2 = 4.5
If x = 3.9, [[x]] = (3 + 4)/2 = 3.5

So, if x = Integer + d (where 'I' is its decimal part whereas 'd' is its decimal part),
then [[x]] = I + 0.5

Statement(1):
[[x]] - x = (I + 0.5) - (I + d) = 0.5 - d
|0.5 - d| will always be between 0 and 1, since d is between 0 and 0.999...
SUFFICIENT.

Statement(2):
0< [[a]] < 1
0 < I + d < 1
Since the decimal value is always between 0 and 1, the integral value of a = 0
So,
0 < a < 1
or
0 <= |a| <= 1
SUFFICIENT.

For statement 1 couldn't x = 0? If x=0 or any other integer say 4 then greatest integer less than or equal to x is x which = 4
and the least integer greater than or equal to x is x which is 4

Would the answer then be B? then you know that it has to be .5
Director
Joined: 14 Dec 2012
Posts: 806
Location: India
Concentration: General Management, Operations
GMAT 1: 700 Q50 V34
GPA: 3.6
Re: Let [[x]] represent the average of the greatest integer less  [#permalink]

### Show Tags

02 Aug 2013, 14:06
MBA2015hopeful wrote:

For statement 1 couldn't x = 0? If x=0 or any other integer say 4 then greatest integer less than or equal to x is x which = 4
and the least integer greater than or equal to x is x which is 4

Would the answer then be B? then you know that it has to be .5

For statement 1 couldn't x = 0? If x=0 or any other integer say 4 then greatest integer less than or equal to x is x which = 4
and the least integer greater than or equal to x is x which is 4

Then the answer for [[x]] is 4 not 4.5. Please explain.==>this is perfectly fine
now when you apply statement 1:[[x]] - x==>4.5 - 4 = 0.5 which is between 0 and 1 (included)

hope it helps
_________________

When you want to succeed as bad as you want to breathe ...then you will be successfull....

GIVE VALUE TO OFFICIAL QUESTIONS...

learn AWA writing techniques while watching video : http://www.gmatprepnow.com/module/gmat-analytical-writing-assessment

Intern
Joined: 14 Jun 2013
Posts: 28
Re: Let ] represent the average of the greatest integer less  [#permalink]

### Show Tags

20 Nov 2014, 05:21
akhileshgupta05 wrote:
Let [[x]] represent the average of the greatest integer less than or equal to x and the least integer greater than or equal to x . Is $$0=< |a| =<1$$?

(1) [[x]] - x = a
(2) $$0< [[a]]<1$$?

I did it this way... dont know whether this is a right method or not..

[[x]]= avg of [(greatest integer less than equal to x)+(smallest integer greater than or equal to x)]
--> greatest integer less than equal to x = number one unit smaller than x...lets say z, similarly
--> smallest int more than or equal to x = number one unit more than x...lets say y
HENCE
z<= x <=y
THESE 3 NUMBERS ARE EITHER 3 CONSECUTIVE NUMBERS OR THEY ARE SAME (as there is <= sign)
so avg of these three numbers, will be X only, even though they are same or consecutive.
Therefore
[[x]] = x
and this helps in solving the question further..
Key is to identify [[x]] = x
Non-Human User
Joined: 09 Sep 2013
Posts: 7755
Re: Let ] represent the average of the greatest integer less  [#permalink]

### Show Tags

20 Dec 2017, 08:49
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: Let ] represent the average of the greatest integer less &nbs [#permalink] 20 Dec 2017, 08:49
Display posts from previous: Sort by

# Events & Promotions

 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®.