In a laboratory, 80 experimental trials were conducted. Each trial was monitored by Sensor… — GMAT

In a laboratory, $80$ experimental trials were conducted. Each trial was monitored by Sensor P, Sensor Q, or both. The average (arithmetic mean) duration of the trials monitored by Sensor P only was $45$ minutes, and the average duration of the trials monitored by Sensor Q only was $60$ minutes. What was the average duration of all $80$ trials?

(1) Exactly $20$ trials were monitored by Sensor P only, and $30$ trials were monitored by Sensor Q only.
(2) The average duration of all trials monitored by Sensor Q was $55$ minutes, and the average duration of the trials monitored by both sensors was $50$ minutes.

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 to answer the question, but neither statement alone is sufficient.

Evaluating Statement (1) gives the counts for all three disjoint regions (

20

P-only,

30

Q-only, and

30

Both), but does not provide the mean duration of the

30

trials in the 'Both' category. Evaluating Statement (2) gives the mean duration of the 'Both' category (

50

minutes) and establishes that

n_{B} = n_{Q}

, but does not give absolute counts. Combining both statements provides all subgroup sizes and their respective means, allowing us to calculate the exact weighted average of

52.5

minutes.

Step-by-Step Solution

Formulate the algebraic expressions for total trials and total duration.

Let

n_{P}

n_{Q}

, and

n_{B}

be the counts for Sensor P only, Sensor Q only, and Both sensors respectively.

n_{P} + n_{Q} + n_{B} = 80

. The total duration is

45 n_{P} + 60 n_{Q} + M_{B} n_{B}

, where

M_{B}

is the mean duration of the trials monitored by both sensors.

Deconstruct the overlapping sets into three mutually exclusive subgroups: P only, Q only, and Both.

Evaluate Statement (1) alone.

n_{P} = 20

and

n_{Q} = 30 \implies n_{B} = 80 - 20 - 30 = 30

. Total duration

= 45(20) + 60(30) + M_{B}(30) = 2700 + 30 M_{B}

Since the mean duration

M_{B}

for trials monitored by both sensors is unknown, Statement (1) is insufficient.

Evaluate Statement (2) alone.

The total group Q consists of Q only (

n_{Q}

, mean

60

) and Both (

n_{B}

, mean

50

). Average for Q is

55 \implies \frac{60 n_{Q} + 50 n_{B}}{n_{Q} + n_{B}} = 55 \implies 5 n_{B} = 5 n_{Q} \implies n_{B} = n_{Q}

Statement (2) proves

n_{B} = n_{Q}

, which gives

n_{P} + 2 n_{Q} = 80

, but without specific numbers for

n_{Q}

n_{P}

, the overall average cannot be uniquely determined. Thus Statement (2) is insufficient.

Evaluate Statements (1) and (2) together.

From (1),

n_{P} = 20

and

n_{Q} = 30

, so

n_{B} = 30

. From (2),

M_{B} = 50

. Total duration

= 45(20) + 60(30) + 50(30) = 900 + 1800 + 1500 = 4200

. Overall average

= \frac{4200}{80} = 52.5

minutes.

Combining both statements yields a unique numeric value for the overall average duration.

Key Concept

Weighted Averages and Overlapping Sets Partitioning

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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