A vessel contains a mixture of milk and water. What is the initial total volume of the mix… — State PSC Exam

A vessel contains a mixture of milk and water. What is the initial total volume of the mixture in the vessel?

Statement I: The initial ratio of milk to water in the vessel is $3:2$ . When $10\text{ liters}$ of water is added to the mixture, the ratio of milk to water becomes $1:1$ .
Statement II: If $15\text{ liters}$ of the original mixture is removed and replaced with $15\text{ liters}$ of pure water, the quantity of milk remaining in the vessel is $27\text{ liters}$ .

Which of the following options correctly describes the sufficiency of the statements to answer the question?

Statement I alone is sufficient, but Statement II alone is not sufficient.Answer
B
Statement II alone is sufficient, but Statement I alone is not sufficient.
C
Either Statement I alone or Statement II alone is sufficient.
D
Neither Statement I nor Statement II is sufficient.
E
Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

Answer

Statement I alone is sufficient to answer the question, but Statement II alone is not sufficient.

Statement I alone gives a complete quantitative relationship: with initial quantities

3x

and

2x

, adding

10\text{ liters}

of water leads to

3x = 2x + 10

, yielding

x = 10

and a unique total initial volume of

50\text{ liters}

. Statement II alone leaves two unknown variables (initial milk fraction and total volume), so it is not sufficient.

Step-by-Step Solution

Evaluate Statement I alone.

Initial volume is calculated as

50\text{ liters}

Let the initial volume of milk be

3x

liters and water be

2x

liters, making the total initial volume

5x

liters. Adding

10\text{ liters}

of water gives the equation

\frac{3x}{2x + 10} = \frac{1}{1}

, which simplifies to

3x = 2x + 10 \implies x = 10

. Thus, total initial volume

5x = 50\text{ liters}

. Hence, Statement I alone is sufficient.

Evaluate Statement II alone.

Insufficient to find total initial volume.

Let the total initial volume be

V

liters and the fraction of milk in the initial mixture be

f

. Removing

15\text{ liters}

of mixture removes

15f

liters of milk. Replacing it with water adds no milk. The remaining milk equation is

f(V - 15) = 27

. Since both

f

and

V

are unknown,

V

cannot be uniquely determined. Hence, Statement II alone is not sufficient.

Key Concept

Data Sufficiency evaluation of mixture ratios and algebraic equations

Estimated Time:1m 30s

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