Two vessels contain milk and water in the ratio 3: 2 and 7: 3. Find the ratio in which the contents of the two vessels have to be mixed to get a new mix ture in which the ratio of milk and water is 2: 1.

Options

A. `2:1`
B. `1:2`
C. `4:1`
D. `1:4`

Answer

To solve the problem of mixing two vessels containing milk and water in the specified ratios, we can follow these steps:

### Step 1: Understand the Ratios in Each Vessel
- Vessel A contains milk and water in the ratio of 3:2.
- Vessel B contains milk and water in the ratio of 7:3.

### Step 2: Calculate the Fraction of Milk and Water in Each Vessel
- For Vessel A:
- Total parts = 3 + 2 = 5
- Milk = 3/5
- Water = 2/5

- For Vessel B:
- Total parts = 7 + 3 = 10
- Milk = 7/10
- Water = 3/10

### Step 3: Set Up the Desired Ratio for the Mixture
- We want the final mixture (Vessel C) to have milk and water in the ratio of 2:1.
- This means:
- Total parts = 2 + 1 = 3
- Milk = 2/3
- Water = 1/3

### Step 4: Use Alligation to Find the Mixing Ratio
- Let the quantity of milk from Vessel A be represented as \( x \) and from Vessel B as \( y \).
- We will set up the alligation method using the fractions of milk:

1. Milk from Vessel A: \( \frac{3}{5} \)
2. Milk from Vessel B: \( \frac{7}{10} \)
3. Milk in the mixture: \( \frac{2}{3} \)

### Step 5: Calculate the Differences
- Find the difference between the milk fractions:
- Difference between Vessel B and the mixture:
\[ \frac{7}{10} - \frac{2}{3} \]
To calculate this, find a common denominator (30):
\[ \frac{7}{10} = \frac{21}{30}, \quad \frac{2}{3} = \frac{20}{30} \]
So,
\[ \frac{21}{30} - \frac{20}{30} = \frac{1}{30} \]

- Difference between the mixture and Vessel A:
\[ \frac{2}{3} - \frac{3}{5} \]
Again, find a common denominator (15):
\[ \frac{2}{3} = \frac{10}{15}, \quad \frac{3}{5} = \frac{9}{15} \]
So,
\[ \frac{10}{15} - \frac{9}{15} = \frac{1}{15} \]

### Step 6: Set Up the Ratio
- The ratio of the differences gives us the ratio in which the two vessels should be mixed:
\[ \text{Ratio} = \frac{1/30}{1/15} = \frac{1}{30} \times \frac{15}{1} = \frac{1}{2} \]
Thus, the ratio of mixing Vessel A and Vessel B is \( 1:2 \).

### Final Answer
The contents of the two vessels should be mixed in the ratio of **1:2**.

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