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# Are all of the numbers in a certain list of 15 numbers equal

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Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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14 Dec 2012, 08:36
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Are all of the numbers in a certain list of 15 numbers equal?

(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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14 Dec 2012, 08:38
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Are all of the numbers in a certain list of 15 numbers equal?

(1) The sum of all the numbers in the list is 60. Clearly insufficient.

(2) The sum of any 3 numbers in the list is 12. Since the sum of ANY 3 numbers is 12 then ALL numbers must equal to 12/3=4, because if not all the numbers equal to 4, then we could pick certain set of 3 numbers so that their sum is not 12. Sufficient.

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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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14 Dec 2012, 08:47
because if not all the numbers equal to 4, then we could pick certain set of 3 numbers so that their sum is not 12. Sufficient...

I didnt get this part bunuel can you please explain this ??
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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14 Dec 2012, 08:51
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ratinarace wrote:
because if not all the numbers equal to 4, then we could pick certain set of 3 numbers so that their sum is not 12. Sufficient...

I didnt get this part bunuel can you please explain this ??

Ask yourself, if not ALL numbers are 4, can the sum of ANY 3 numbers in the list be 12?
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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23 Dec 2012, 06:54
Bunuel wrote:
ratinarace wrote:
because if not all the numbers equal to 4, then we could pick certain set of 3 numbers so that their sum is not 12. Sufficient...

I didnt get this part bunuel can you please explain this ??

Ask yourself, if not ALL numbers are 4, can the sum of ANY 3 numbers in the list be 12?

Bunnel - Cant we hypothesize that there are only three elements such as :{10,1,1} ,(2,8,2}...
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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23 Dec 2012, 06:57
1
vishu1414 wrote:
Bunuel wrote:
ratinarace wrote:
because if not all the numbers equal to 4, then we could pick certain set of 3 numbers so that their sum is not 12. Sufficient...

I didnt get this part bunuel can you please explain this ??

Ask yourself, if not ALL numbers are 4, can the sum of ANY 3 numbers in the list be 12?

Bunnel - Cant we hypothesize that there are only three elements such as :{10,1,1} ,(2,8,2}...

Stem says: Are all of the numbers in a certain list of 15 numbers equal?

So, we know that there are 15 elements in the set.
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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26 Dec 2012, 02:44
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Dear Bunnel,

I still can't understand. Would you please explain more about "list of 15 numbers"?
1. Why can't we hypothesize that there are only three elements such as :{10,1,1} ,(2,8,2}... , and the remaining elements such as {4,4,5,5,6,7......}
2. If the three elements must be 4, why the remaining elements have to be equal as well?
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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26 Dec 2012, 03:14
4
violachiang wrote:
Dear Bunnel,

I still can't understand. Would you please explain more about "list of 15 numbers"?
1. Why can't we hypothesize that there are only three elements such as :{10,1,1} ,(2,8,2}... , and the remaining elements such as {4,4,5,5,6,7......}
2. If the three elements must be 4, why the remaining elements have to be equal as well?

What are you hypothesizing about? We are told that there are 15 elements. Ask yourself, if the set is {10, 1, 1, 2, 8, 2, ...}, is the sum of ANY 3 numbers in the set 12? No! 10+2+8=18.

Also, from (2) we have that ALL elements must be 4, not just 3 of them.
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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26 Dec 2012, 03:34
2
I finally understand what's my problem!
I didn't understand the sentence 2 clearly. It is "The sum of any 3 numbers in the list is 12. "
Thanks a lot, Bunnel!
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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29 Apr 2013, 09:32
2
Reducing shortcut:

After reading the question stem and noting that this is question is about lists/statistics, note that on all the numbers in the question stem are a multiples of 3. So reduce down and re-word the question:

"Are all of the numbers in a certain list of 5 numbers equal?" 15/3 = 5

1) "The sum of all the numbers in the list is 20" 60/3 = 20

Easily insufficient, as there are many ways to add up to 20.

If you wanna test numbers:

Our list:
_ _ _ _ _ = 20
4 4 4 4 4 = 20 yes, all numbers are equal
4 4 4 5 1 = 20 no, all numbers are not equal

2) "The sum of any 1 number in the list is 4?" 3/3 = 1 12/3 = 4

Sufficient, it should become apparent to you that all the numbers are now equal:

Our list: 4 4 4 4 4 = 20

How to get this:

Average = Sum/# of terms
4 = Sum/5
20 = Sum
"Any 1" number is 4
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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29 May 2014, 08:45
Hi Bunuel,
Even if st2 was "sum of any two number is 12", it would be still be sufficient, right?
Thanks!
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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29 May 2014, 08:54
1
MensaNumber wrote:
Hi Bunuel,
Even if st2 was "sum of any two number is 12", it would be still be sufficient, right?
Thanks!

_____________
Right.
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Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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14 Apr 2015, 13:57
2
OG explanation for Choice 2 is unnecessarily complicated, but here is my easy clear explanation:

Choice (2): The sum of any 3 numbers in the list is 12.

OK. Let's name the 15 numbers by letters, a, b, c, d, ...

Consider a, b, and c, three arbitrary numbers from the list; we know from (2) that a + b + c= 12 (I)

Now consider a, b and d; we know from (2) that a + b + d= 12 (II)

From (I) and (II) we have: a + b + c = a+ b + d ---> c=d

We can again consider a + b + d, a + b + e , ... and prove that d=e and hence c = d = e

In the similar way it can be proved that a = b = c = d = e = f = ...
------------------------------------------------------------------------------------------------------------

Suppose in (2) it were said 'The sum of any 4 (or 5, 7, etc.) numbers in the list is y'; then we could similarly prove that every two numbers of the list were equal.
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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14 Apr 2015, 23:13
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Hi All,

This DS question is really about considering the "possibilities" and making sure that you're thorough with your thinking.

We're told that there is a group of 15 numbers. We're asked if they're all equal. This is a YES/NO question.

Fact 1: The sum of the numbers is 60

IF.....
We have fifteen 4s, then the answer to the question is YES.

IF....
We have ANY OTHER option (e.g. fourteen 3s and one 18), then the answer to the question is NO.
Fact 1 is INSUFFICIENT

Fact 2: The sum of ANY 3 numbers in the list is 12.

With THIS information, we know that all the numbers MUST be 4s. Here's why:

With fifteen 4s, we know that selecting ANY 3 of them will give us a sum of 12. If we change EVEN 1 of those numbers to something else though, then there's no way to GUARANTEE that we get a total of 12 from any 3.

For example, if we have fourteen 4s and one 5. It's possible that we could pick 3 numbers and get 4+4+5 = 13, which is NOT a sum of 12. We're told that picking ANY 3 numbers gets us a sum of 12 though, so this serves as proof that no other option exists. Therefore, all fifteen numbers MUST be 4s and the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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15 Apr 2015, 07:42
1
EMPOWERgmatRichC wrote:
Hi All,

This DS question is really about considering the "possibilities" and making sure that you're thorough with your thinking.

We're told that there is a group of 15 numbers. We're asked if they're all equal. This is a YES/NO question.

Fact 1: The sum of the numbers is 60

IF.....
We have fifteen 4s, then the answer to the question is YES.

IF....
We have ANY OTHER option (e.g. fourteen 3s and one 18), then the answer to the question is NO.
Fact 1 is INSUFFICIENT

Fact 2: The sum of ANY 3 numbers in the list is 12.

With THIS information, we know that all the numbers MUST be 4s. Here's why:

With fifteen 4s, we know that selecting ANY 3 of them will give us a sum of 12. If we change EVEN 1 of those numbers to something else though, then there's no way to GUARANTEE that we get a total of 12 from any 3.

For example, if we have fourteen 4s and one 5. It's possible that we could pick 3 numbers and get 4+4+5 = 13, which is NOT a sum of 12. We're told that picking ANY 3 numbers gets us a sum of 12 though, so this serves as proof that no other option exists. Therefore, all fifteen numbers MUST be 4s and the answer to the question is ALWAYS YES.
Fact 2 is SUFFICIENT

GMAT assassins aren't born, they're made,
Rich

In analyzing the fact 2, while you have provided some intuition, but this cannot be indeed called a proof, a formal one.

The proof is, however, much easier than what you have explained:

For every x and y of this list, consider two other numbers from the list (let's name these latter two by a, b):

We know from fact 2 that: x + a + b= 12 (I) and y + a + b=12 (II)

From (I) and (II) we have : x = y

This means that any two arbitrarily chosen numbers of the list are equal, meaning that all the numbers are equal.
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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26 Nov 2015, 14:06
Bunuel, In fact st2 could have been sum of any 2,3,4,....14 is equal to a certain number and still would have been enough, right?
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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26 Nov 2015, 16:23
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Hi MensaNumber,

To fit all of the 'design elements' of the question, Fact 2 could have been written in a number of different 'versions':

For example...
The sum of ANY 2 numbers in the list is 8.
The sum of ANY 4 numbers in the list is 16.
The sum of ANY 5 numbers in the list is 20.
The sum of ANY 6 numbers in the list is 24.
Etc.

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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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01 Sep 2017, 15:02
Top Contributor
Are all of the numbers in a certain list of 15 numbers equal?

(1) The sum of all the numbers in the list is 60.
(2) The sum of any 3 numbers in the list is 12.

Target question: Are all 15 numbers equal?

Statement 1: The sum of all the numbers in the list is 60.
There are several possible scenarios that satisfy this statement. Here are two.
Case a: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, in which case all of the numbers are equal
Case b: numbers are: {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 7}, in which case all of the numbers are not equal
Statement 1 is NOT SUFFICIENT

Statement 2: The sum of any 3 numbers in the list is 12.
This is a very powerful statement, because it tells us that all of the numbers in the set are equal.
Let's let a,b,c and d be four of the 15 numbers in the set.
We know that a + b + c = 12
Notice that if I replace ANY of these three values (a,b or c) with d, the sum must still be 12.
This tells us that a, b and c must all equal d.
I can use a similar approach to show that e, f and g must also equal d.
In fact, I can show that ALL of the numbers in the set must equal d, which means all of the numbers in the set must be equal.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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18 Nov 2017, 12:52
Bunuel
For statement 2, 5+4+3 gives 12 as a result. 10+1+1 also.
Then how can we conclude that it has to only be 4,4,4.

Thanks

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Re: Are all of the numbers in a certain list of 15 numbers equal [#permalink]

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19 Nov 2017, 03:15
Bunuel
For statement 2, 5+4+3 gives 12 as a result. 10+1+1 also.
Then how can we conclude that it has to only be 4,4,4.

Thanks

Posted from my mobile device

We are told that there are 15 elements not 3. (2) says: The sum of ANY 3 numbers in the list is 12. If not ALL numbers are 4, the sum of ANY 3 numbers in the list won't be 12.
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Re: Are all of the numbers in a certain list of 15 numbers equal   [#permalink] 19 Nov 2017, 03:15

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