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 consisting half milk and half water?

Options

A. `8:3`
B. `7:5`
C. `4:3`
D. `2:3`

Answer

To solve the problem step by step, 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 consists of half milk and half water.

### Step 1: Understand the ratios in vessels A and B
- In vessel A, the ratio of milk to water is 4:3. This means:
- Milk in A = 4 parts
- Water in A = 3 parts
- Total parts in A = 4 + 3 = 7 parts

- In vessel B, the ratio of milk to water is 2:3. This means:
- Milk in B = 2 parts
- Water in B = 3 parts
- Total parts in B = 2 + 3 = 5 parts

### Step 2: Calculate the fractions of milk and water in each vessel
- In vessel A:
- Fraction of milk = 4/7
- Fraction of water = 3/7

- In vessel B:
- Fraction of milk = 2/5
- Fraction of water = 3/5

### Step 3: Set up the desired mixture in vessel C
We want vessel C to have a mixture that is half milk and half water, which can be represented as:
- Fraction of milk in C = 1/2
- Fraction of water in C = 1/2

### Step 4: Use the method of allegations
To find the ratio in which the two vessels should be mixed, we can use the method of allegations.

1. **Calculate the difference for water:**
- Water in A = 3/7
- Water in B = 3/5
- Water in C = 1/2

Using the formula:
\[ \text{Difference for A} = \text{Water in A} - \text{Water in C} = \frac{3}{7} - \frac{1}{2} = \frac{6 - 7}{14} = -\frac{1}{14} \]
(We take the absolute value for the purpose of ratio.)

\[ \text{Difference for B} = \text{Water in C} - \text{Water in B} = \frac{1}{2} - \frac{3}{5} = \frac{5 - 6}{10} = -\frac{1}{10} \]
(Again, we take the absolute value.)

2. **Set up the ratio:**
The ratio of the differences gives us the ratio in which to mix A and B:
\[ \text{Ratio} = \frac{1/10}{1/14} = \frac{14}{10} = \frac{7}{5} \]

### Final Answer
The liquids in both vessels A and B should be mixed in the ratio of **7:5** to obtain a new mixture in vessel C consisting of half milk and half water.