The milk and water in two vessels A and B are in the ratio 4:3 and 2: 3 respectively. In what ratio, the liquids in both the vessels be mixed to obtain a new mixture in vessel C containing half milk and half water?

Options

A. (a) `7:5`
B. (b) `5:2`
C. (c) `3:11`
D. (d) `1:2`

Explanation

To solve the problem, we need to find the ratio in which the liquids from vessels A and B should be mixed to obtain a new mixture in vessel C that contains half milk and half water.

### Step-by-Step Solution:

1. **Identify the Ratios of Milk and Water in Each Vessel:**
- In vessel A, the ratio of milk to water is 4:3.
- In vessel B, the ratio of milk to water is 2:3.

2. **Calculate the Proportion of Milk and Water in Each Vessel:**
- For vessel A:
- Total parts = 4 (milk) + 3 (water) = 7
- Proportion of milk in A = 4/7
- Proportion of water in A = 3/7
- For vessel B:
- Total parts = 2 (milk) + 3 (water) = 5
- Proportion of milk in B = 2/5
- Proportion of water in B = 3/5

3. **Set Up the Equation for the Mixture in Vessel C:**
- We want the mixture in vessel C to have half milk and half water, which can be expressed as:
- Proportion of milk in C = 1/2
- Proportion of water in C = 1/2

4. **Use the Alligation Method:**
- We can set up the alligation to find the ratio of the two mixtures:
- Milk in A: 4/7
- Milk in B: 2/5
- Milk in C: 1/2

5. **Calculate the Differences:**
- Difference between milk in A and milk in C:
\[ \text{Difference A} = \frac{4}{7} - \frac{1}{2} \]
To calculate this, find a common denominator (14):
\[ \frac{4}{7} = \frac{8}{14}, \quad \frac{1}{2} = \frac{7}{14} \quad \Rightarrow \quad \text{Difference A} = \frac{8}{14} - \frac{7}{14} = \frac{1}{14} \]

- Difference between milk in B and milk in C:
\[ \text{Difference B} = \frac{1}{2} - \frac{2}{5} \]
Again, find a common denominator (10):
\[ \frac{1}{2} = \frac{5}{10}, \quad \frac{2}{5} = \frac{4}{10} \quad \Rightarrow \quad \text{Difference B} = \frac{5}{10} - \frac{4}{10} = \frac{1}{10} \]

6. **Set Up the Ratio:**
- The ratio of the quantities of liquids from vessels A and B to be mixed is given by the inverses of the differences:
\[ \text{Ratio} = \text{Difference B} : \text{Difference A} = \frac{1}{10} : \frac{1}{14} \]

7. **Simplify the Ratio:**
- To simplify, we can multiply both sides by the least common multiple of the denominators (70):
\[ \text{Ratio} = 70 \cdot \frac{1}{10} : 70 \cdot \frac{1}{14} = 7 : 5 \]

### Final Answer:
The ratio in which the liquids in both vessels should be mixed to obtain the desired mixture in vessel C is **7:5**.