Bunuel wrote:

In a product test of a common cold remedy, x percent of the patients tested experienced side effects from the use of the drug and y percent experienced relief of cold symptoms. What percent of the patients tested experienced both side effects and relief of cold symptoms?

(1) Of the 1,000 patients tested, 15 percent experienced neither side effects nor relief of cold symptoms.

(2) Of the patients tested, 30 percent experienced relief of cold symptoms without side effects.

This is an overlapping set question. A great way to solve this problem is to set up a table with two main categories: experienced side effects and experienced relief from cold symptoms. More specifically, our table will be labeled as follows:

1) Side Effects

2) No Side Effects

3) Relief

4) No Relief

We are given that x percent of the patients tested experienced side effects from the use of the drug and y percent experienced relief from cold symptoms.

We need to determine what percentage of the patients tested experienced both side effects and relief from cold symptoms.

Let’s fill all of this into a table where each entry is the percentage of the total patients tested.

Statement One Alone:Of the 1,000 patients tested, 15 percent experienced neither side effects nor relief of cold symptoms.

With the information from statement one, we can put 15 in the “No Side Effects and No Relief” cell and, subsequently, the values (in terms of x and y) in the other three cells of our table.

We see that the percentage of patients tested who experienced side effects and relief from cold symptoms is (x + y - 85) percent. However, since we know neither the value of x nor the value of y, we do not have enough information to determine the percentage of people who both experienced relief and had side effects. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:Of the patients tested, 30 percent experienced relief of cold symptoms without side effects.

With the information from statement two, we can enter 30 in the “Relief and No Side Effects” cell and, subsequently, the values (in terms of x and y) in the other three cells of our table.

We see that the percentage of patients tested who experienced both side effects and relief from cold symptoms is (y - 30) percent. However, since we do not know the value of y, we do not have enough information to determine the percentage of people who both experienced relief and had side effects. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:Using the information from statements one and two, we have the following:

As we can see, from the “Side Effects” column, we have (y - 30) + (85 - y) = x, and from the “No side Effects” column, we have 30 + 15 = 100 - x. From either equation, we can determine that x = 55. However, since we still don’t know the value of y, we do not have enough information to determine the percentage of people who both experienced relief and had side effects.

Answer: E

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