Question: Dataset A consists of 10 terms, each of which is a reciprocal of a prime number, is the median of the dataset less than \(\frac{1}{5}\)?

1. Reciprocal of the median is a prime number
2. The product of any two terms of the set is a terminating decimal

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.

Answer:
Solution with Explanation:
Approach Solution (1):

S1: Reciprocal of the median is a prime number. If all the terms equal \(\frac{1}{2}\), then the median = \(\frac{1}{2}\) and the answer is no but if all the terms equal \(\frac{1}{7}\), then the median = \(\frac{1}{7}\) and the answer is yes
Not sufficient

S2: The product of any two terms of the set is a terminating decimal. This statement implies that the set must consists of \(\frac{1}{2}\) or / and \(\frac{1}{5}\). Thus the median could be \(\frac{1}{2}\), \(\frac{1}{5} or \frac{\frac{1}{5} + \frac{1}{2}}{2} = \frac{7}{20}\)

None of the possible values is less than \(\frac{1}{5}\)

Sufficient

Correct Option: B

Approach Solution (2):

Set A consists of 10 terms. The median of a set with even number of terms is the average of two middle terms, when arranged in ascending/descending order.

If two middle terms are \(\frac{1}{5}\), then the median is simply \(\frac{1}{5}\)

If two middle terms are \(\frac{1}{2}\) , then the median is simply \(\frac{1}{2}\)

If two middle terms are \(\frac{1}{5}\) and \(\frac{1}{2}\), then the median is \(\frac{\frac{1}{5} + \frac{1}{2}}{2} = \frac{7}{20}\)

Correct Option: B

Approach Solution (3):

Statement 1: The median of the numbers is 30
There are several possible sets that satisfy this condition. Here are two:
Case a: set A = {1/7, 1/7, 1/7,...1/7} in which case the median = 1/7, so the median IS less than 1/5
Case b: set A = {1/2, 1/2, 1/2,...1/2} in which case the median = 1/2, so the median is NOT less than 1/5
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The product of any two terms of the set is a terminating decimal
There's a nice rule that says something like this:
If a/b results in a terminating decimal, then the denominator, b, MUST be the product of 2's and 5's only!
So, for example, if b = 20, the fraction a/b will result in a terminating decimal. The same holds true for other values of b such as 4, 5, 25, 40, 2, 8, and so on.

So, statement 1 tells us that set A must consist of 1/2's and 1/5's ONLY.
Since set A has an EVEN number of terms, the median will be the AVERAGE of the two middlemost terms.
Since the terms must be 1/2's and 1/5's ONLY, there are only three possible cases.
case a: the two middlemost terms are 1/2 and 1/2, in which case the median is 1/2, which means the median is NOT less than 1/5
case b: the two middlemost terms are 1/5 and 1/5, in which case the median is 1/5, which means the median is NOT less than 1/5
case c: the two middlemost terms are 1/5 and 1/2, in which case the median is 7/20, which means the median is NOT less than 1/5
In all three cases, the median is NOT less than 1/5
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Correct Option: B

“Dataset A consists of 10 terms, each of which is a reciprocal of a prime number, is the median of the dataset less than \(\frac{1}{5}\)?”- 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.

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