Explanation

In this question, 30 out of 100 ratings of product were negative, and 45 out of 100 ratings of service were negative, and we want to know the overlap of those ratings--the number of unique people who rated both negatively. The answer could be as high as 30 (in the case of maximum overlap of the negative ratings) or as low as 0 (in the case of no overlap of the negative ratings), based on what we know so far. On to the statements, separately first.

Statement (1) gives us the count of people who fit in neither group. Since we are getting "neither" (N), and we want "both" (B), we can use the overlapping sets formula, which you can recall or derive from a Venn diagram:



Note that this formula doesn't describe the whole table, but rather just the "Negative" column of the table. From the table, we know that . From the data statement, we now know , so we can solve for B. Statement (1) is therefore sufficient.

Statement (2) gives us some information about the "Positive" column. Is it relevant to the number who rated both negative? It's relevant, because it allows us to compute the number of unique people in the "Positive" column. But there is still a range of possibilities. For example, the "Neutral" column could represent 15 unique people or, say, 20 unique people, with a corresponding difference in the number of unique people in the "Negative" column. Statement (2) does not allow us to answer the question definitively, so it's insufficient.

The correct answer is (A).