The questions given below contain two statements numbered I and II giving certain data. You have to decide whether the data given in the statements are sufficient to answer the question. Give answer
A vessel contains 'X Litres of mixture of milk and water. The ratio of milk to water in the vessel is 5:3. What is the numerical value of 'X'?
i If32 litres of mixture is taken out and 2 litres of wateris added, the percentage of water in the resultant mixture will become 40%.
II If.4 litres of milk and 6 litres of water is added to the mixture, the ratio of milk to water in the vessel will become 3:2.

Options

A. if statement I alone is sufficient but statement II alone is not sufficient.
B. if statement II alone is sufficient but statement I alone is not sufficient.
C. if either statement I or II is sufficient.
D. if both statement I and II together are not suficient.

Answer

To solve the problem, we need to analyze the two statements given and determine if they provide sufficient information to find the numerical value of 'X', the total volume of the mixture in the vessel.

### Step-by-Step Solution:

1. **Understanding the Initial Condition:**
- We have a mixture of milk and water in the ratio of 5:3.
- Let the quantity of milk be \(5A\) and the quantity of water be \(3A\).
- Therefore, the total volume of the mixture \(X = 5A + 3A = 8A\).

2. **Analyzing Statement I:**
- The statement says that if 32 liters of the mixture is taken out and 2 liters of water is added, the percentage of water in the resultant mixture will become 40%.
- From the initial mixture of 8A liters, when 32 liters is taken out, the amount of milk taken out is:
\[ \text{Milk taken out} = \frac{5}{8} \times 32 = 20 \text{ liters} \]
- The amount of water taken out is:
\[ \text{Water taken out} = \frac{3}{8} \times 32 = 12 \text{ liters} \]
- After removing 32 liters, the remaining quantities are:
- Milk left = \(5A - 20\)
- Water left = \(3A - 12\)
- Then, 2 liters of water is added:
- New quantity of water = \(3A - 12 + 2 = 3A - 10\)
- The total mixture now is:
\[ \text{Total mixture} = (5A - 20) + (3A - 10) = 8A - 30 \]
- According to the statement, the percentage of water in the new mixture is 40%:
\[ \frac{3A - 10}{8A - 30} \times 100 = 40 \]
- This gives us one equation with one unknown (A), which can be solved to find A, and subsequently, X.

3. **Analyzing Statement II:**
- The second statement states that if 4 liters of milk and 6 liters of water are added, the ratio of milk to water becomes 3:2.
- After adding, the new quantities will be:
- Milk = \(5A + 4\)
- Water = \(3A + 6\)
- The ratio of milk to water is given as:
\[ \frac{5A + 4}{3A + 6} = \frac{3}{2} \]
- Cross-multiplying gives us another equation with one unknown (A), which can also be solved to find A, and subsequently, X.

4. **Conclusion:**
- Both statements I and II provide sufficient information independently to find the value of A, and thus the value of X.
- Therefore, the answer is that either statement alone is sufficient to answer the question.

### Final Answer:
The answer is option 3: Each statement alone is sufficient.