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Data Sufficiency Problem 156

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Data Sufficiency Problem 156

by acarrazzone » Tue Dec 15, 2015 1:38 pm
The explanation for the following problem was a bit confusing and I was hoping someone can further explain the answer.

Are all the numbers in a certain list of 15 numbers equal?
(1) The sum of all numbers in the list is 60
(2) The sum of any 3 numbers in the list is 12.

The answer is B. I understand why statement 1 is not sufficient, but can someone further explain how to solve using statement 2?
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by Brent@GMATPrepNow » Tue Dec 15, 2015 1:43 pm
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

Answer = B

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by [email protected] » Tue Dec 15, 2015 3:23 pm
Hi acarrazzone,

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

Final Answer: B

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by Amrabdelnaby » Sat Jan 02, 2016 2:59 pm
Hello,

First, We need to know basically if are numbers in the set are the equal.
If all numbers in the set are equal then any number is equal to the mean.

for example consider this set {6,6,6}

this set has 3 sixes, the mean of them is (6+6+6)/3 which is also 6.

now lets examine each statement.

statement one tells us the sum of the elements within the set. ok, does this tell us anything about the nature of the numbers in the set? are the positive for example or negative? is there a range between them? no.. nothing. so statement one alone is insufficient.

lets look at statement two:

the sum of any 3 numbers in the set is 12.... ok what does this mean?

this means that if you pick any 3 numbers at random in the set their sum will be 12.... so in order to do so all three numbers that we pick must add up to 12, and the only way to do so is by having 3 equal numbers that are a factor of 12. ie: when u divide 12 by 3 (number of the elements we picked) you get 4.

so basically this statement is telling you any three numbers we pick are {4,4,4}

hence we know that this set has elements of 4 only because if it had any other number beside four it was not going to be the case that EVERYTIME we pick 3 elements they will add up to 12.

I hope this helps.
acarrazzone wrote:The explanation for the following problem was a bit confusing and I was hoping someone can further explain the answer.

Are all the numbers in a certain list of 15 numbers equal?
(1) The sum of all numbers in the list is 60
(2) The sum of any 3 numbers in the list is 12.

The answer is B. I understand why statement 1 is not sufficient, but can someone further explain how to solve using statement 2?
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