A logistics center operates two automated sorting divisions: Division P and Division Q. Ye… — GMAT

A logistics center operates two automated sorting divisions: Division $P$ and Division $Q$ . Yesterday, Division $P$ had an error rate of $x\%$ of the packages it processed, and Division $Q$ had an error rate of $y\%$ of the packages it processed. What was the overall package error rate for the two divisions combined yesterday?

(1) Division $P$ processed $50\%$ more packages yesterday than Division $Q$ processed yesterday, and Division $P$ 's error rate was $2.0\%$ .
(2) Yesterday, Division $Q$ processed $40\%$ of the total packages processed by both divisions combined, and Division $Q$ 's error rate was $5.0\%$ .

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 answer is the option stating that both statements together are sufficient, but neither alone is sufficient. Statement (1) provides the relative volume weights and Division

P

's error rate, but lacks Division

Q

's error rate. Statement (2) provides the relative volume weights and Division

Q

's error rate, but lacks Division

P

's error rate. Combining both statements supplies all necessary values (

x = 2.0\%

y = 5.0\%

, and volume ratio

60:40

) to uniquely calculate the combined weighted error rate of

3.2\%

Step-by-Step Solution

Formulate the algebraic target for the overall combined error rate.

Let

N_P

and

N_Q

represent the package volumes processed by Division

P

and Division

Q

, respectively. The overall error rate

E

is given by the weighted average equation:

E = \frac{x\cdot N_P + y\cdot N_Q}{N_P + N_Q} = x\left(\frac{N_P}{N_P + N_Q}\right) + y\left(\frac{N_Q}{N_P + N_Q}\right)

. To find

E

, we need

x

y

, and the relative weight ratio

\frac{N_P}{N_Q}

(or the fraction of total volume contributed by each division).

Rephrasing the Data Sufficiency question stem shows that absolute counts for

N_P

and

N_Q

are unnecessary; only the error rates

x

and

y

and their relative proportions are required.

Evaluate Statement (1) independently.

Statement (1) states that

N_P = 1.5 N_Q

, which means

\frac{N_P}{N_Q} = \frac{3}{2}

and the relative weights are

\frac{N_P}{N_P + N_Q} = 0.60

and

\frac{N_Q}{N_P + N_Q} = 0.40

. It also gives

x = 2.0\%

. However, no information is provided about Division

Q

's error rate (

y

). Thus,

E = 0.60(2.0\%) + 0.40(y\%)

, which varies depending on

y

. Statement (1) alone is INSUFFICIENT.

Without

y

, a unique numerical value for

E

cannot be calculated.

Evaluate Statement (2) independently.

Statement (2) states that Division

Q

processed

40\%

of the total packages, meaning

\frac{N_Q}{N_P + N_Q} = 0.40

and

\frac{N_P}{N_P + N_Q} = 0.60

. It also provides

y = 5.0\%

. However, no information is provided about Division

P

's error rate (

x

). Thus,

E = 0.60(x\%) + 0.40(5.0\%)

, which varies depending on

x

. Statement (2) alone is INSUFFICIENT.

Without

x

, a unique numerical value for

E

cannot be calculated.

Evaluate Statements (1) and (2) together.

Combining both statements gives

x = 2.0\%

y = 5.0\%

, and consistent relative weights (

\frac{N_P}{N_P + N_Q} = 0.60

and

\frac{N_Q}{N_P + N_Q} = 0.40

). The overall combined error rate can be computed directly:

E = 0.60(2.0\%) + 0.40(5.0\%) = 1.2\% + 2.0\% = 3.2\%

. A unique value is obtained. Both statements together are SUFFICIENT.

All required variables (

x

y

, and relative volume weighting) are known when combining both statements.

Key Concept

Weighted Averages and Ratio Sufficiency in Data Sufficiency

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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