If Set S Consists of Even Number of Integer, Is the Median GMAT Data Sufficiency

Question: If set S consists of even number of integer, is the median of set S negative?

1. Exactly half of all the elements of the set S are positive.
2. The largest negative element of set S is -1.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If set S consists of even number of integer, is the median of set S negative?”– is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach Solution 1:

Notice that since the set consists of even number of integers then the median will be the average of the two middle elements.

1. Exactly half of all the elements of set S are positive. Either all other elements are negative for example S can be {-30, -20, -10, 1, 2, 3} or all other elements but 0 are negative, for example set S can be {-3, -2, -1, 0, 1, 2, 3, 4}.
   Hence this statement is not sufficient
2. The largest negative element of set S is -1
   Hence this statement is not sufficient.

(1) + (2) we have that exactly half of all the elements are positive and the largest negative element is -1.

If zero is in S, the median equals \(\frac{0+positive}{2}\)= positive.

If zero is NOT in S, the median equals \(\frac{-1+positive integer}{2}\geq0\) = non negative

So the answer to the question whether the median is negative is NO.

Hence this statement is sufficient.

Correct Answer: C

Approach Solution 2:

The question asks: is the median of set S negative? When combined we get a NO answer to the question the median is NOT negative. For your example the answer remains the same the median = \(\frac{-1+2}{2}=\frac{1}{2}\) , so it’s not negative. All possible cases give a NO answer, so the statements combined are sufficient.

Correct Answer: C

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