M14-05

Question

If set  S  consists of even number of integers, is the median of set  S  negative?

(1) Exactly half of the elements in  S  are positive.
 
(2) The greatest negative element in  S  is -1.

Bunuel · Accepted Answer

Official Solution:

If set  S  consists of even number of integers, is the median of set  S  negative? 
 
First, note that since the set contains an even number of integers, the median will be the average of the two middle elements.
 
(1) Exactly half of all elements of set  S  are positive. 
 
The above implies that the remaining half of the elements in the set are composed of either negative elements only, or negative elements and 0. In the first scenario, the median would be  \frac{negative + positive}{2} , which could be positive, negative or zero depending on the specific values of these two middle elements. However, in the second scenario, the median would be  \frac{0 + positive}{2}=positive . This statement alone is not sufficient.
 
(2) The largest negative element of set  S  is -1. This statement alone is not sufficient.
 
(1)+(2) From the above, we know that exactly half of the elements in  S  are positive, and the largest negative element is -1. There are two possible scenarios for the median value of  S :
 
If  S  contains 0, then the median value would be  \frac{0 + positive}{2}=positive .
 
If  S  does not contain 0, then the median value would be  \frac{-1+	ext{positive integer}}{2}=	ext{0 or positive} .
 
Therefore, in either case, the answer to the question of whether the median of  S  is negative is NO, making the statements together sufficient.

Answer: C

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Fdambro294 · Answer

the set contains an even number of integers

statement 2 tells us that the greatest negative integer = --1

if n = 2: {-1 , 0}

the median is negative

if n = 4: {-1 , 2 , 5 , 7}

the median is positive

statement 2 is not sufficient

together:

statement 1 tells us that exactly half of the integers are positive
statement 2 tells us that the greatest negative integer is --1

if we try to contain the maximum amount of negative integers (-1s) in the set, the other half of the integers must be positive.
since the smallest positive integer is +1, the minimum we can make the median is when we have a set such as:

{-1 , -1 , 1 , 1}

which would still give us a NO answer to the question: Is the median negative?

*C*

note: the key case to not forget when analyzing statement 2 alone is the case in which you have only 2 elements in the set.

in all other cases, it will not be possible to have a negative median.

M14-05

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