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Manager  Joined: 02 Dec 2012
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Are all of the numbers in a certain list of 15 numbers equal  [#permalink]

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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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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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GMAT 1: 730 Q49 V41 Are all of the numbers in a certain list of 15 numbers equal  [#permalink]

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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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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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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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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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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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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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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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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.

There are a few ways to look at this. One is to reverse the problem: say they aren't all equal. Write the set in increasing order: {a, b, c, ..., m, n, o}, and while some of these might be equal, we must have a < o. Well clearly then the sum of the three smallest numbers is less than the sum of the three largest, (a+b+c < m+n+o), so the sum of any three numbers in the list isn't always the same. So the only way S2 can be true is if all the numbers are equal.
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Manager  Joined: 25 Jul 2012
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Re: Are all of the numbers in a certain list of 15 numbers equal  [#permalink]

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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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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Are all of the numbers in a certain list of 15 numbers equal  [#permalink]

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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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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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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Are all of the numbers in a certain list of 15 numbers equal  [#permalink]

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

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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]

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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.

We need to determine whether all of the numbers in the certain list of 15 numbers are equal.

Statement One Alone:

The sum of all the numbers in the list is 60.

Using the information in statement one, we see that all the numbers could be equal or could not be equal. In one scenario, all the numbers could equal 4, and in another scenario 13 of 15 the numbers could equal 4 and the last two numbers could be 6 and 2. In either case, the sum would be 60. Thus, statement one alone is not sufficient to answer the question.
Statement Two Alone:

The sum of any 3 numbers in the list is 12.

Let’s create variables for all 15 numbers in our list.

A, B, C, D, E, F, G, H, I, J, K, L, M, N, O

Using the information in statement two, we see that, regardless of which 3 numbers we select, they MUST sum to 12.

For example, A + B + C = 12 or A + B + N = 12 or A + C + N = 12.

We see that the only way this is possible is if all the numbers are the same value. Statement two alone is sufficient to answer the question.

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

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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.

I think I've posted different solutions to this question elsewhere, but one way to look at Statement 2:

let a, b, c and d be four random numbers from the list. From Statement 2, since the sum of *any* three numbers is 12, we know

a + b + c = 12
d + b + c = 12

and if you subtract the second equation from the first, you find that a - d = 0, so a = d. Since a and d are just randomly chosen numbers from the list, we could do the same thing for any of the numbers in our list, and so they all need to be equal. Since Statement 1 is not sufficient, the answer is B.
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Are all of the numbers in a certain list of 15 numbers equal  [#permalink]

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I still didn't get you....

There are 15 different numbers, but we took only 4 for our example.

what if a + b+ c = 12
and d + e+ f = 12
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