A logistics center processes packages using two sorting machines, Machine X and Machine Y,… — GMAT

A logistics center processes packages using two sorting machines, Machine X and Machine Y, each operating continuously at its own constant hourly rate. On a given operational shift, Machine X processed packages for $t_x$ hours and Machine Y processed packages for $t_y$ hours. What percentage of the total number of packages processed by both machines combined during the shift was processed by Machine X?

(1) If Machine X had operated for $20\%$ more time and Machine Y had operated for $20\%$ less time, Machine X would have processed $50\%$ more packages than Machine Y.

(2) The ratio of the hourly processing rate of Machine X to the hourly processing rate of Machine Y was $3:2$ , and the total operating time of both machines combined was $10$ hours.

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

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (1) alone is sufficient because translating the percentage modifications into an equation gives

1.2 P_x = 1.5 \times (0.8 P_y)

, which simplifies to

1.2 P_x = 1.2 P_y

, or

P_x = P_y

. This proves Machine X processed exactly

50\%

of the total packages. Statement (2) alone is not sufficient because knowing

t_x + t_y = 10

and

r_x / r_y = 3/2

allows infinitely many different package output ratios depending on how the

10

hours are split between the two machines.

Step-by-Step Solution

Define target variable algebraically from question stem

Let

r_x

and

r_y

be the constant hourly processing rates of Machine X and Machine Y, respectively. The total packages processed by Machine X is

P_x = r_x t_x

, and by Machine Y is

P_y = r_y t_y

. The target percentage processed by Machine X is

\frac{P_x}{P_x + P_y} \times 100\% = \frac{1}{1 + \frac{P_y}{P_x}} \times 100\%

. Finding the ratio

\frac{P_x}{P_y}

is necessary and sufficient.

Rephrasing the question stem shows that finding the single ratio of total packages

P_x / P_y

is sufficient to answer the question.

Evaluate Statement (1)

Under the hypothetical scenario in Statement (1), the new operating time for Machine X is

1.2 t_x

and for Machine Y is

0.8 t_y

. The new package totals are

r_x (1.2 t_x) = 1.2 P_x

and

r_y (0.8 t_y) = 0.8 P_y

. The statement specifies that

1.2 P_x = 1.5 \times (0.8 P_y)

. Simplifying the right-hand side:

1.5 \times 0.8 = 1.2

, so

1.2 P_x = 1.2 P_y \implies P_x = P_y

. Thus,

\frac{P_x}{P_y} = 1

, which means Machine X processed

\frac{P_x}{P_x + P_x} = 50\%

of the total packages.

Algebraic simplification yields a unique value for the target percentage, making Statement (1) sufficient.

Evaluate Statement (2)

Statement (2) gives

\frac{r_x}{r_y} = \frac{3}{2}

and

t_x + t_y = 10

. The ratio of total packages processed is

\frac{P_x}{P_y} = \left(\frac{r_x}{r_y}\right) \left(\frac{t_x}{t_y}\right) = \frac{3}{2} \left(\frac{t_x}{t_y}\right)

. Since

t_x

and

t_y

can take any positive real values summing to

10

(e.g.,

t_x = 2, t_y = 8 \implies \frac{P_x}{P_y} = \frac{3}{8}

vs.

t_x = 6, t_y = 4 \implies \frac{P_x}{P_y} = \frac{9}{4}

), the percentage varies.

Without individual values for

t_x

and

t_y

, multiple target percentages are possible, so Statement (2) is not sufficient.

Key Concept

Question Stem Rephrasing and Algebraic Simplification of Combined Rates

Estimated Time:2m 30s

Question

Answer

Step-by-Step Solution

Key Concept

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