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Source: — Data Sufficiency |
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by GMATGuruNY » Thu Aug 20, 2015 3:22 am
tapanmittal wrote:144. 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.
Statement 1:
Case 1: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
Here, all of the numbers are equal.
Case 2: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 5
Here, all of the numbers are NOT equal.
INSUFFICIENT.

Statement 2:
Case 1: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
Here, no matter which 3 numbers are chosen, the sum = 4+4+4 = 12.

Case 3: 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1?
Not viable:
If the last 3 numbers are chosen, the sum = 4+4+1 = 9, violating the constraint that the sum of ANY 3 NUMBERS must be 12.

As Case 3 illustrates -- for the sum of ANY 3 NUMBERS to be 12 -- ALL of the numbers MUST BE 4.
Thus, all of the numbers must be EQUAL.
SUFFICIENT.

The correct answer is B.
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by Brent@GMATPrepNow » Thu Aug 20, 2015 5:58 am
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] » Thu Aug 20, 2015 9:40 am
Hi tapanmittal,

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 Max@Math Revolution » Fri Aug 21, 2015 8:16 pm
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and equations ensures a solution.



144. 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.

In the original condition we have 15 variables, and we need 15 equations to match the number of variables thus E is likely the answer. Using both 1) & 2) together, all the 15 variables are all 4, thus the solution is yes and the answer is C.

In integer and statistic problems, variable approach methods and common mistake types relation require us to solve the problem using 1) and 2) separately. In common mistake type 4(A) it says "if it's easily C, consider A, B" and with 2) we find out that all 15 variables are 4, and the solution is still yes and also sufficient. Thus the answer is b.


If you know our own innovative logics to find the answer, you don't need to actually solve the problem.
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l The one-and-only World's First Variable Approach for DS and IVY Approach for PS that allow anyone to easily solve GMAT math questions.

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