Of the 58 Patients of Vertigo Hospital, 45 have Arachnophobia GMAT Data Sufficiency

Question: Of the 58 patients of Vertigo Hospital, 45 have arachnophobia. How many of the patients have acrophobia?

(1) The number of patients of Vertigo Hospital who have both arachnophobia and acrophobia is the same as the number of patients who have neither arachnophobia nor acrophobia.
(2) 32 patients of Vertigo Hospital have arachnophobia but not acrophobia.

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.

Correct Answer: A
Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:

• Of the 58 patients of Vertigo Hospital, 45 have arachnophobia.

Find out:

• The number of patients having acrophobia.

Statement one alone:
No. of patients of Vertigo Hospital having both arachnophobia and acrophobia is identical to the no. of patients having neither arachnophobia nor acrophobia.

We can solve the question by using a double set matrix:

Therefore, the number of patients who have acrophobia = 58 – 45 = 13.
Hence, statement one alone is sufficient to find the number of patients having acrophobia.

Statement two alone:
32 patients of Vertigo Hospital have arachnophobia but not acrophobia.

Clearly insufficient to find the number of patients having acrophobia.

Approach Solution 2:
The problem statement informs that:
Given:

• Of the 58 patients of Vertigo Hospital, 45 have arachnophobia.

Find out:

• The number of patients having acrophobia.

Let’s solve the problem by using the Venn diagram approach.

Here, No. of patients having arachnophobia = 45
Let no. of patients having acrophobia = Y.

Statement one alone:
No. of patients of Vertigo Hospital having both arachnophobia and acrophobia is identical to the no. of patients having neither arachnophobia nor acrophobia

We need to calculate Y which is equal to B+X

From the given conditions of the question, we can say:

(45-X) + X + B + X = 58
=> B + X = 58 - 45
=> B + X = 13
=> Y = 13

Therefore, the number of patients having acrophobia = 13

Statement two alone:
32 patients of Vertigo Hospital have arachnophobia but not acrophobia.

This statement contradicts the given information in the question.
Hence, statement two alone is not sufficient to find the number of patients having acrophobia.

“Of the 58 patients of Vertigo Hospital, 45 have arachnophobia”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. The GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions include a problem statement that is 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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