A logistics center uses two sorting systems, Line 1 and Line 2, to process package invento… — GMAT

A logistics center uses two sorting systems, Line 1 and Line 2, to process package inventory. Line 1 operates at a constant rate of $r_1$ packages per minute, and Line 2 operates at a constant rate of $r_2$ packages per minute. During a testing run, Line 1 operated for $t_1$ minutes and Line 2 operated for $t_2$ minutes, sorting a total of $P$ packages. Was the combined average sorting rate during the test run—defined as the total packages sorted divided by total machine-minutes worked, $\frac{P}{t_1 + t_2}$ —greater than $40$ packages per minute?

(1) Line 1's sorting rate $r_1$ was $50\%$ greater than Line 2's sorting rate $r_2$ , and Line 1 operated for a duration $t_1$ that was $50\%$ longer than Line 2's operating duration $t_2$ .
(2) If Line 1 and Line 2 were to operate simultaneously for $1$ hour, they would sort a combined total of $3{}000$ packages.

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 to determine that the combined average rate is 26 packages per minute, which definitively answers the question with a 'No'. Neither statement alone is sufficient.

The correct option is the one stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) simplifies the weighted average rate to

1.3 r_2

, which is insufficient by itself because

r_2

is unknown. Statement (2) provides

r_1 + r_2 = 50

, which is insufficient alone because individual rates and time proportions are unknown. Combining both statements allows us to solve for

r_2 = 20

and compute the exact weighted average rate of

26

packages per minute. Since

26

is not greater than

40

, we obtain a definitive 'No' answer, establishing sufficiency.

Adım Adım Çözüm

Rephrase the target question algebraically.

The combined average sorting rate is

\text{Average Rate} = \frac{P}{t_1 + t_2} = \frac{r_1 t_1 + r_2 t_2}{t_1 + t_2}

. The target question asks: Is

\frac{r_1 t_1 + r_2 t_2}{t_1 + t_2} > 40

Establishing the target formula in terms of

r_1, r_2, t_1,

and

t_2

allows direct evaluation of each statement.

Evaluate Statement (1) independently.

Statement (1) gives

r_1 = 1.5 r_2

and

t_1 = 1.5 t_2

. Substituting these into the average rate formula gives

\frac{(1.5 r_2)(1.5 t_2) + r_2 t_2}{1.5 t_2 + t_2} = \frac{2.25 r_2 t_2 + r_2 t_2}{2.5 t_2} = \frac{3.25 r_2 t_2}{2.5 t_2} = 1.3 r_2

Since the value of

r_2

is unknown, we cannot determine whether

1.3 r_2 > 40

. Statement (1) alone is INSUFFICIENT.

Evaluate Statement (2) independently.

Operating simultaneously for

1

hour (

60

minutes) produces

3{}000

packages:

60(r_1 + r_2) = 3000 \implies r_1 + r_2 = 50

packages per minute.

Knowing

r_1 + r_2 = 50

does not fix the ratio of durations

t_1 / t_2

or individual rates

r_1, r_2

. The weighted average could be anywhere between

r_1

and

r_2

. Statement (2) alone is INSUFFICIENT.

Evaluate Statements (1) and (2) together.

From Statement (1),

r_1 = 1.5 r_2

. Substituting into Statement (2)'s equation

r_1 + r_2 = 50

yields

1.5 r_2 + r_2 = 50 \implies 2.5 r_2 = 50 \implies r_2 = 20

. Consequently,

r_1 = 30

. Substituting

r_2 = 20

into the expression from Statement (1) gives an average rate of

1.3 \times 20 = 26

packages per minute.

Since

26

is not greater than

40

, we can answer the question with a definitive 'No'. A definitive 'No' means the combined statements are SUFFICIENT.

Anahtar Kavram

Weighted average rate simplification and Yes/No sufficiency determination in Data Sufficiency.

Tahmini Süre:2m 30s

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