Consecutiveness of n

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sidceg: Senior | Next Rank: 100 Posts; Posts: 89; Joined: Fri Jun 01, 2012 12:22 am; Location: Bangalore, India; Thanked: 6 times; Followed by:2 members

Consecutiveness of n

by sidceg » Mon Oct 07, 2013 5:17 pm

Hi! This was from one of the MGMAT CATs. Can some one please explain?

A list contains n distinct integers. Are all n integers consecutive?

(1) The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed.

(2) The positive difference between any two numbers in the list is always less than n.

OA is D

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Source: Beat The GMAT — Data Sufficiency | Mon Oct 07, 2013 5:17 pm

theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pm; Thanked: 448 times; Followed by:34 members; GMAT Score:650

by theCodeToGMAT » Mon Oct 07, 2013 7:24 pm

N = A, B,C,D,E,F

Statement 1:
(B+C+D+E+F)/5 = 1 + (A+B+C+D+E)/5
B+C+D+E+F = 5 + A+B+C+D+E
F-A = 5
Only consecutive number series is possible to give such a result
Hence, consecutive.
SUFFICIENT

Statement 2:
F-A < 6
Since, numbers are different..
Only consecutive number series is possible to give such a result
SUFFICIENT

Answer [spoiler]{D}[/spoiler]

R A H U L

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 am; Thanked: 2256 times; Followed by:1535 members; GMAT Score:800

by lunarpower » Tue Oct 08, 2013 3:26 am

^^ The post above shows that the statements are sufficient for n = 6, but they don't necessarily represent all values of n. (There's no reason to assume that n is 6.)

If you're going to take the "plug in numbers" approach, then, good -- it's an effective approach -- but you should definitely try more than one value of n!

Ron has been teaching various standardized tests for 20 years.

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pm; Thanked: 448 times; Followed by:34 members; GMAT Score:650

by theCodeToGMAT » Tue Oct 08, 2013 3:42 am

lunarpower wrote:^^ The post above shows that the statements are sufficient for n = 6, but they don't necessarily represent all values of n. (There's no reason to assume that n is 6.)

If you're going to take the "plug in numbers" approach, then, good -- it's an effective approach -- but you should definitely try more than one value of n!

Ron, from the result of statement one

F-A = 5 [A,B,C,D,E & F are integer values]

it's pretty much clear that RHS would always be one less than the value of "n" I could think of..

Hence, trying multiple values of "n" would be merely waste of time that I could have saved for next questions..That's why i din't try more values.

R A H U L

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 am; Thanked: 2256 times; Followed by:1535 members; GMAT Score:800

by lunarpower » Tue Oct 08, 2013 3:47 am

theCodeToGMAT wrote:Ron, from the result of statement one

F-A = 5 [A,B,C,D,E & F are integer values]

it's pretty much clear that RHS would always be one less than the value of "n" I could think of..

Hence, trying multiple values of "n" would be merely waste of time that I could have saved for next questions..That's why i din't try more values. :) ..

Fair enough. If you see the seeds of a strong pattern, then, great. Nicely done.

I guess my point is that, since you're going to the trouble of posting these things on a public forum, you may as well lay out that part of the reasoning more explicitly.
E.g., "I can tell that the same thing will happen for other n's." Or something like that.

In other words, even if your solution is awesome, there's little point in posting it without that kind of explanatory detail.
(Maybe I misunderstand the point of view of people on the forum, though. For instance, there are tons and tons of posts on every thread that say things like "I choose B", with no explanation whatsoever. Such posts add zero value to the discussion -- and are frustrating to scroll through, if there are lots of them -- but people still post them.)

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theCodeToGMAT: Legendary Member; Posts: 1556; Joined: Tue Aug 14, 2012 11:18 pm; Thanked: 448 times; Followed by:34 members; GMAT Score:650

by theCodeToGMAT » Tue Oct 08, 2013 3:54 am

Thanks Ron. Yeah, I agree with you .. I will try my best to post "Sound" solutions

R A H U L

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sidceg: Senior | Next Rank: 100 Posts; Posts: 89; Joined: Fri Jun 01, 2012 12:22 am; Location: Bangalore, India; Thanked: 6 times; Followed by:2 members

by sidceg » Wed Oct 16, 2013 3:21 am

Hi Ron and theCodeToGMAT,

I am extremely sorry I lost track of this thread while I was checking my other posts and posts by others in this forum.

I kinda got the first statement. But still confused with the second. What does 'positive' difference mean in the first place? Can you please explain statement II once again?

Again, apologies for missing this thread.

Thanks

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 am; Thanked: 2256 times; Followed by:1535 members; GMAT Score:800

by lunarpower » Thu Oct 17, 2013 1:43 am

sidceg wrote:Hi Ron and theCodeToGMAT,

I am extremely sorry I lost track of this thread while I was checking my other posts and posts by others in this forum.

I kinda got the first statement. But still confused with the second. What does 'positive' difference mean in the first place? Can you please explain statement II once again?

Again, apologies for missing this thread.

Thanks

"Positive difference" is exactly what it says. It's the difference between two numbers, expressed as a positive number.
E.g., the positive difference between 3 and 7 is 4. The positive difference between 7 and 3 is also 4.

For statement 2, the only thing you really need to think about is the difference between the biggest and smallest values in the set. (If you don't see why that is so, think about it for a bit.)
Try making any set, at all, that doesn't consist entirely of consecutive integers. See whether you can get that difference to be less than N.

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sidceg: Senior | Next Rank: 100 Posts; Posts: 89; Joined: Fri Jun 01, 2012 12:22 am; Location: Bangalore, India; Thanked: 6 times; Followed by:2 members

by sidceg » Thu Oct 17, 2013 2:16 am

lunarpower wrote:
sidceg wrote:Hi Ron and theCodeToGMAT,

I am extremely sorry I lost track of this thread while I was checking my other posts and posts by others in this forum.

I kinda got the first statement. But still confused with the second. What does 'positive' difference mean in the first place? Can you please explain statement II once again?

Again, apologies for missing this thread.

Thanks
"Positive difference" is exactly what it says. It's the difference between two numbers, expressed as a positive number.
E.g., the positive difference between 3 and 7 is 4. The positive difference between 7 and 3 is also 4.

For statement 2, the only thing you really need to think about is the difference between the biggest and smallest values in the set. (If you don't see why that is so, think about it for a bit.)
Try making any set, at all, that doesn't consist entirely of consecutive integers. See whether you can get that difference to be less than N.

I tried the following cases.

1,2,3: 3-2<3 --> Yes
4,5,6,7: 7-4<4 --> Yes
-3,-2,-1,0,1: 1-(-3)<5 --> Yes

But what about [-5,0,5,10]? Are these integers consecutive?

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by lunarpower » Thu Oct 17, 2013 2:28 am

"Consecutive" means "next to each other in sequence, without skipping anything".
The difference between any two consecutive integers is 1.

5, 10, 15, 20, ... would be consecutive multiples of 5. But they're not consecutive integers, since there are other integers between them.

--

More importantly, those values are irrelevant to the problem at hand, because they don't satisfy statement 2.

Statement 2 says that the difference between any 2 integers in the set is LESS than N.
Try getting that to happen with anything other than a set of consecutive integers. Not gonna happen.

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sidceg: Senior | Next Rank: 100 Posts; Posts: 89; Joined: Fri Jun 01, 2012 12:22 am; Location: Bangalore, India; Thanked: 6 times; Followed by:2 members

by sidceg » Thu Oct 17, 2013 3:12 am

lunarpower wrote:"Consecutive" means "next to each other in sequence, without skipping anything".
The difference between any two consecutive integers is 1.

5, 10, 15, 20, ... would be consecutive multiples of 5. But they're not consecutive integers, since there are other integers between them.

Okay I often got confused between the consecutive numbers as numbers spaced equally.

lunarpower wrote:More importantly, those values are irrelevant to the problem at hand, because they don't satisfy statement 2.

Statement 2 says that the difference between any 2 integers in the set is LESS than N.
Try getting that to happen with anything other than a set of consecutive integers. Not gonna happen.

Since I had considered consecutive numbers as numbers equally spaced, I tried the above numbers also which yielded a NO. So with all other cases YES and this one case NO, I concluded that statement II is not sufficient.

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 am; Thanked: 2256 times; Followed by:1535 members; GMAT Score:800

by lunarpower » Thu Oct 17, 2013 3:22 am

Ah, I see. But, no, that's not what "consecutive integers" means.

There's a different name for that sort of thing -- I think it's "arithmetic progression" or "arithmetic sequence", or something like that -- but you won't need to know that name for the gmat exam.

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sidceg: Senior | Next Rank: 100 Posts; Posts: 89; Joined: Fri Jun 01, 2012 12:22 am; Location: Bangalore, India; Thanked: 6 times; Followed by:2 members

by sidceg » Thu Oct 17, 2013 3:36 am

Alright! Thanks Ron

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PerfectScores: Newbie | Next Rank: 10 Posts; Posts: 7; Joined: Sat Jan 04, 2014 3:09 am; Location: Online

by PerfectScores » Sat Jan 11, 2014 3:41 am

sidceg wrote:Hi! This was from one of the MGMAT CATs. Can some one please explain?

A list contains n distinct integers. Are all n integers consecutive?

(1) The average (arithmetic mean) of the list with the lowest number removed is 1 more than the average (arithmetic mean) of the list with the highest number removed.

(2) The positive difference between any two numbers in the list is always less than n.

OA is D

Statement I is sufficient

Let us say n = 4 and the numbers are 4, 5, 6 and 7

If we remove 4 then the average is 6
If we remove 7 then the average is 5 which satisfies with the statement

Now lets go algebra:

If the numbers are {x, x+1, x+2.....x+n-1} with average (2x + n - 1)/2 (First + last)/2

If we remove x then the average will become (x + 1 + x + n - 1)/2 = (2x + n)/2 increasing by 1/2
If we remove x + n - 1 the average will become (x + x + n -2)/2 = (2x + n -2)/2 decreasing by 1/2

Hence the difference will always be the 1 between them.

Statement II is sufficient:

If the numbers are {x, x+1, x+2.....x+n-1}

If we subtract the first and the last we will get = |x + n - 1 - x| = n - 1 which is always less than n. Now is there any other set which can have that. Since the members of the set are distinct it is not possible to have that.

Answer is D.

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lunarpower: GMAT Instructor; Posts: 3380; Joined: Mon Mar 03, 2008 1:20 am; Thanked: 2256 times; Followed by:1535 members; GMAT Score:800

by lunarpower » Sun Jan 12, 2014 8:51 am

PerfectScores wrote:Statement I is sufficient

Let us say n = 4 and the numbers are 4, 5, 6 and 7

If we remove 4 then the average is 6
If we remove 7 then the average is 5 which satisfies with the statement

Now lets go algebra:

If the numbers are {x, x+1, x+2.....x+n-1} with average (2x + n - 1)/2 (First + last)/2

If we remove x then the average will become (x + 1 + x + n - 1)/2 = (2x + n)/2 increasing by 1/2
If we remove x + n - 1 the average will become (x + x + n -2)/2 = (2x + n -2)/2 decreasing by 1/2

Hence the difference will always be the 1 between them.

^^ It appears here that you're starting with a set of consecutive integers, and trying to prove statement 1. That's not the way it works, although you'll get lucky with the answer here.

If you want to take an approach like this one, then you must also prove that starting with any other set (i.e., a set that doesn't consist entirely of consecutive integers) won't give a difference of 1.

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