A logistics company analyzed the operational downtime of a fleet of 80 delivery trucks ove… — GMAT

A logistics company analyzed the operational downtime of a fleet of $80$ delivery trucks over a one-month period. Each truck in the fleet completed at least one of two specialized maintenance programs: Program A or Program B. The arithmetic mean downtime for trucks that completed only Program A was $12$ hours, and the arithmetic mean downtime for trucks that completed only Program B was $18$ hours. What was the average downtime per truck, in hours, for all $80$ trucks in the fleet?

(1) Exactly $50$ trucks completed Program A, and exactly $45$ trucks completed Program B.
(2) The total downtime of all trucks in the fleet that completed Program B was $765$ hours.

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.Cevap
D
EACH statement ALONE is sufficient.
E
Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

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

The correct response identifies that both statements together provide complementary pieces of required information: the first statement determines the number of trucks in the 'only A' group (

35

trucks), while the second statement provides the combined downtime of all trucks in the 'Program B' group (

765

hours). Together, they uniquely specify the fleet's total downtime as

12(35) + 765 = 1185

hours, giving a single average of

14.8125

hours.

Adım Adım Çözüm

Define set variables and express the target fleet average in terms of known and unknown quantities.

Let

n_A

be the number of trucks completing only Program A,

n_B

be the number of trucks completing only Program B, and

n_{AB}

be the number of trucks completing both programs. The total number of trucks is

n_A + n_B + n_{AB} = 80

. The total downtime of the fleet is

T_{total} = (12 \cdot n_A) + T_B

, where

T_B

is the total downtime of all trucks that completed Program B (which comprises trucks completing only B and trucks completing both A and B).

Partitioning the fleet into trucks completing only Program A and trucks completing Program B allows us to rephrase the total downtime simply as

12 n_A + T_B

Evaluate Statement (1) independently.

Statement (1) states that

n_A + n_{AB} = 50

and

n_B + n_{AB} = 45

. Since

(n_A + n_{AB}) + (n_B + n_{AB}) - n_{AB} = 80

, we get

50 + 45 - n_{AB} = 80 \Rightarrow n_{AB} = 15

. This yields

n_A = 35

and

n_B = 30

. However, we do not know

T_B

or the downtime of the

15

trucks in both programs, so the fleet average cannot be calculated. Statement (1) alone is NOT sufficient.

Knowing set counts alone does not supply the necessary downtime data for the overlapping region.

Evaluate Statement (2) independently.

Statement (2) gives

T_B = 765

hours. Substituting this into our total downtime expression yields

T_{total} = 12 n_A + 765

. Since

n_A

(the number of trucks completing only Program A) is unknown,

T_{total}

cannot be uniquely calculated. Statement (2) alone is NOT sufficient.

Without knowing

n_A

, the contribution of trucks completing only Program A to total downtime cannot be determined.

Evaluate Statements (1) and (2) together.

From Statement (1),

n_A = 35

. From Statement (2),

T_B = 765

. Thus,

T_{total} = 12(35) + 765 = 420 + 765 = 1185

hours. The average downtime per truck is

\frac{1185}{80} = 14.8125

hours, which is a unique numerical value. Both statements together are SUFFICIENT.

Combining

n_A = 35

with

T_B = 765

uniquely determines the fleet's total downtime and overall average.

Anahtar Kavram

Overlapping set partitioning and weighted averages in Data Sufficiency

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