A pharmaceutical laboratory manufactures a specialized compound by blending two liquid pre… — GMAT

A pharmaceutical laboratory manufactures a specialized compound by blending two liquid preparations, Solution $X$ and Solution $Y$ . Solution $X$ contains Active Ingredient $P$ and water in the ratio $3:2$ by weight. Solution $Y$ contains Active Ingredient $P$ and water in the ratio $1:4$ by weight. A master batch is created by mixing $x$ grams of Solution $X$ with $y$ grams of Solution $Y$ , where $x > 0$ and $y > 0$ . Is the percentage of Active Ingredient $P$ in the master batch greater than $40\%$ by weight?

(1) $3x - 4y > 0$
(2) After adding $50$ grams of pure water to the master batch, Active Ingredient $P$ accounts for less than $\frac{1}{3}$ of the total weight of the resulting mixture.

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.

The correct response identifies that Statement (1) alone is sufficient while Statement (2) alone is not. By rephrasing the question stem, the target condition 'Is the active ingredient concentration greater than 40%?' simplifies directly to the inequality condition

x > y

. Statement (1) guarantees that

x > \frac{4}{3}y

, which directly implies

x > y

since

y > 0

, providing a definitive 'Yes'. Statement (2) simplifies to

2x - y < 125

, which is satisfied by parameter pairs where

x > y

as well as pairs where

x < y

, making it insufficient.

Step-by-Step Solution

Rephrase the question stem target algebraically.

Solution X is

\frac{3}{3+2} = \frac{3}{5} = 0.60

(or

60\%

) Ingredient P by weight. Solution Y is

\frac{1}{1+4} = \frac{1}{5} = 0.20

(or

20\%

) Ingredient P by weight. The total weight of Ingredient P in the master batch is

0.6x + 0.2y

, and the total weight of the batch is

x + y

. The question asks whether

\frac{0.6x + 0.2y}{x + y} > 0.40

. Since

x > 0

and

y > 0

, multiplying across by

(x + y)

gives

0.6x + 0.2y > 0.4x + 0.4y \implies 0.2x > 0.2y \implies x > y

. Thus, the rephrased question is: 'Is

x > y

Simplifying the question stem before evaluating statements prevents unnecessary system solving and clarifies the exact threshold needed for sufficiency.

Evaluate Statement (1):

3x - 4y > 0

Rearranging

3x - 4y > 0

gives

3x > 4y \implies x > \frac{4}{3}y

. Because

y > 0

\frac{4}{3}y > y

. Therefore, if

x > \frac{4}{3}y

, it must strictly be true that

x > y

. This yields a definitive 'Yes' to the rephrased question. Hence, Statement (1) ALONE is sufficient.

Since statement 1 establishes a lower bound for x relative to y that is strictly greater than 1y, it answers the question 'Is x > y?' definitively.

Evaluate Statement (2): Adding 50 grams of water results in P accounting for less than

\frac{1}{3}

of total weight.

Total weight of P remains

0.6x + 0.2y

. The new total weight is

x + y + 50

. The statement gives

\frac{0.6x + 0.2y}{x + y + 50} < \frac{1}{3}

. Multiplying by

3(x + y + 50)

yields

1.8x + 0.6y < x + y + 50 \implies 0.8x - 0.4y < 50 \implies 2x - y < 125

. Testing values:
- Case A: Let

x = 10, y = 50

. Then

2(10) - 50 = -30 < 125

holds. Here

x < y

, so the answer is 'No'.
- Case B: Let

x = 100, y = 90

. Then

2(100) - 90 = 110 < 125

holds. Here

x > y

, so the answer is 'Yes'.
Since Statement (2) permits both 'Yes' and 'No' outcomes, Statement (2) ALONE is not sufficient.

An inequality involving absolute scale (50 grams) fails to determine a purely proportional relationship between x and y without additional constraints on total batch size.

Key Concept

Question Stem Simplification and Mixture Ratio Inequalities

Question

Answer

Step-by-Step Solution

Key Concept

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