A medical research trial evaluated 100 patients, each of whom experienced side effect A, s… — GMAT

A medical research trial evaluated $100$ patients, each of whom experienced side effect A, side effect B, or both. Exactly $65$ patients experienced side effect A, and exactly $55$ patients experienced side effect B. What was the average (arithmetic mean) duration of side effect A, in days, among all patients who experienced side effect A?

(1) The average duration of side effect A for patients who experienced ONLY side effect A was $8$ days.
(2) The average duration of side effect A for patients who experienced BOTH side effects was $12$ days.

A
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

The correct choice indicates that both statements combined provide sufficient data while neither is sufficient on its own. Using the standard inclusion-exclusion principle for two sets,

100 = 65 + 55 - N(\text{Both})

, giving exactly

20

patients with both side effects and

45

patients with only side effect A. Statement (1) provides the sum of durations for the

45

patients (

45 \times 8 = 360

), and Statement (2) provides the sum of durations for the

20

patients (

20 \times 12 = 240

). Neither statement alone gives both sums, but together they yield a total sum of

600

days for

65

patients, resulting in a single definitive mean of

\frac{600}{65}

days.

Step-by-Step Solution

Rephrase the question stem using overlapping sets principles.

Let

N(A) = 65

N(B) = 55

, and total patients

N(A \cup B) = 100

. Using the set formula

N(A \cup B) = N(A) + N(B) - N(A \cap B)

, we get

100 = 65 + 55 - N(A \cap B)

, so

N(A \cap B) = 20

Determining the exact count of patients in the overlap region (

A \cap B

) and the exclusive region (

A \text{ only}

) is necessary to set up the weighted average formula.

Calculate the subgroup sizes for patients experiencing side effect A.

Patients with ONLY side effect A =

N(A) - N(A \cap B) = 65 - 20 = 45

. Patients with BOTH side effects =

20

. Total patients with side effect A =

65

The total duration of side effect A across all

65

patients is the sum of durations from the

45

'only A' patients and the

20

'both' patients.

Evaluate Statement (1) independently.

Statement (1) states that the mean for the

45

'only A' patients is

8

days. The total duration for this group is

45 \times 8 = 360

days. However, the duration for the

20

'both' patients is unknown.

Without the total duration for the overlap group, the overall mean cannot be determined. Thus, Statement (1) alone is NOT sufficient.

Evaluate Statement (2) independently.

Statement (2) states that the mean for the

20

'both' patients is

12

days. The total duration for this group is

20 \times 12 = 240

days. However, the duration for the

45

'only A' patients is unknown.

Without the total duration for the exclusive A group, the overall mean cannot be determined. Thus, Statement (2) alone is NOT sufficient.

Evaluate Statement (1) and Statement (2) together.

Combining both statements, the total duration for all

65

patients with side effect A is

360 + 240 = 600

days. The overall average duration is

\frac{600}{65} = \frac{120}{13}

days.

A unique numerical value is obtained for the target question, so both statements together are sufficient.

Key Concept

Combining overlapping set subgroup sizes with weighted arithmetic mean formulas to evaluate Data Sufficiency statements.

Estimated Time:2m 0s

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