The milk and water in two vessels are in the ratio of 3 : 1 and 7 : 11 respectively. In what ratio should the liquid in both the vessels be mixed to obtain a new mixture containing half milk and half water?

Options

A. `5:7`
B. `4:9`
C. `1:1`
D. `4:7`

Answer

To solve the problem of mixing the liquids from two vessels with given ratios of milk and water to obtain a new mixture containing equal parts of milk and water, we can follow these steps:

### Step 1: Understand the Ratios
- In the first vessel, the ratio of milk to water is 3:1. This means for every 4 parts, 3 parts are milk and 1 part is water.
- In the second vessel, the ratio of milk to water is 7:11. This means for every 18 parts, 7 parts are milk and 11 parts are water.

### Step 2: Calculate the Proportions of Milk and Water
- For the first vessel:
- Total parts = 3 (milk) + 1 (water) = 4 parts
- Proportion of milk = 3/4
- Proportion of water = 1/4

- For the second vessel:
- Total parts = 7 (milk) + 11 (water) = 18 parts
- Proportion of milk = 7/18
- Proportion of water = 11/18

### Step 3: Set Up the Mixture Requirement
We want to create a new mixture that has equal parts of milk and water, which corresponds to a ratio of 1:1.

### Step 4: Use Allegation Method
To find the ratio in which the two mixtures should be combined, we can use the allegation method.

1. **Milk Content in Each Vessel:**
- From Vessel 1: \( \frac{3}{4} \) (or 0.75)
- From Vessel 2: \( \frac{7}{18} \) (approximately 0.3889)

2. **Desired Milk Content:**
- For a 1:1 ratio, the desired milk content is \( \frac{1}{2} \) (or 0.5).

3. **Set Up the Allegation:**
- The difference between the milk content of Vessel 1 and the desired content:
\[ 0.75 - 0.5 = 0.25 \]
- The difference between the milk content of Vessel 2 and the desired content:
\[ 0.5 - 0.3889 \approx 0.1111 \]

### Step 5: Calculate the Ratio
The ratio of the two vessels can be determined by the differences calculated:
- The ratio of the amounts from Vessel 1 to Vessel 2 is:
\[ \text{Ratio} = \frac{0.25}{0.1111} \approx 2.25 \text{ or } \frac{9}{4} \]

### Final Answer
The ratio in which the liquids from the two vessels should be mixed is approximately \( 9:4 \).

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