BTGmoderatorDC wrote:If S is a set of distinct positive multiples of 5, what is the smallest element of S?

(1) The mean of S and the median of S both equal 20.
(2) The largest element of S is 40.

OA C

Source: Princeton Review

Let's take each statement one by one.

(1) The mean of S and the median of S both equal 20.

Case 1: Say S: {5, 20, 35}; we see that both mean and median are 20 and the smallest element is 5.
Case 2: Say S: {10, 20, 30}; we see that both mean and median are 20 and the smallest element is 10.

No unique answer. Insufficient.

(2) The largest element of S is 40.

Certainly insufficient. The smallest element can be anyone among 5, 10, 15, 20, 25, 30 and 35. Insufficient.

(1) and (2) together

From (1), we know that the median is 20, which is a multiple of 5; thus, there must be odd no. of elements in Set S as the elements are distinct.

So, taking (1) and (2) together...

Case 1: Say S: {5, 20, 40}; we see that though the median is 20, the mean is not 20. So, this is not a valid case.
Case 2: Say S: {10, 20, 40}; we see that though the median is 20, the mean is not 20. So, this is also not a valid case.
Case 3: Say S: {15, 20, 40}; we see that though the median is 20, the mean is not 20. So, this is also not a valid case.

From these cases, we deduce that there must not be 3 elements in Set S; thus, there can be 5 elements.

Case 4: Say S: {5, 10, 20, 25, 40}; we see that both mean and median are 20 and the smallest element is 5. This is a valid case.

We cannot exclude 5 and include 15 (10 greater than 5) since by doing so, in order to maintain mean = 20, we will have to exclude 25 and include 15 (10 less than (25); however the median will not be 20 then; moreover, the elements are distinct, too.

Taking 7 elements...

Case 5: Say S: {5, 10, 15, 20, 25, 25, 40}; we see that both mean and median are 20 and the smallest element is 5. However, the elements are not distinct. This is not a valid case. We cannot exclude 25 and include 30 or 35, else mean would be greater than 20.

So, Case 4 is the only possibility. The smallest element is 5. Sufficient

The correct answer: C

Hope this helps!

-Jay
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