D01-23

Official Solution:


List A consists of 10 terms, each of which is a reciprocal of a prime number, is the median of the list less than 1/5?

(1) The reciprocal of the median is a prime number.

If all the terms in the list are 1/2, then the median is 1/2, which is not less than 1/5. On the other hand, if all the terms are 1/7, then the median is 1/7, which is less than 1/5. Not sufficient.

(2) The product of any two terms of the list is a terminating decimal.

Since the terms in the list are reciprocals of prime numbers, the product of any two terms will be a reduced fraction \(\frac{1}{p*q}\), where \(p\) and \(q\) are prime numbers. For a reduced fraction, in its decimal form, to have a finite number of digits after the decimal point, it must have only 2's and/or 5's in its denominator. Therefore, the list can only consist of 1/2's, 1/5's, or a combination of 1/2's and 1/5's. It cannot include the reciprocal of any other prime, such as 1/3, because no pair with 1/3 will produce a product that is a terminating decimal.

If the two middle terms are both 1/2, then the median is 1/2, which is NOT less than 1/5.

If the two middle terms are both 1/5, then the median is 1/5, which is NOT less than 1/5.

Finally, if the two middle terms are 1/5 and 1/2, then the median is (1/5 + 1/2)/2 = 7/20, which is also NOT less than 1/5.

Since none of the possible medians are less than 1/5, the answer to the question is no. Therefore, statement (2) alone is sufficient to answer the question.


Answer: B