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questions on averages and sum

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by abhasjha » Thu Nov 21, 2013 8:18 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.
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Thu Nov 21, 2013 8:25 am
abhasjha wrote: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

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Mathsbuddy » Fri Nov 22, 2013 2:42 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.

Consider the obvious case where it is true:

60/15 = 4 -> 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
We see that 4 4 4 -> 3 * 4 = 12 for any 3 values

Let's now try to change it:
4 4 4 4 4 4 4 4 4 4 4 4 4 (4+k)(4-k)
Here is a case where statement (2) is not always met
e.g. 4 + 4 + (4+k) is not necessarily 12

Therefore it is necessary for all numbers to be equal.
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by Mathsbuddy » Fri Nov 22, 2013 2:48 am
Or even more simply:

If we positively change one value, at least one other value must be negatively changed.
Hence the sum of ANY three cannot be guaranteed to be constant.
Therefore to sustain statement (2) without changing total=60 then they must be all equal.
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