Vessels A and B contain mixtures of milk and water in the ratios 4:5 and 5: 1 respectively. In what ratio should quantities of mixture be taken from A and B to form a mixture in which milk to water is in the ratio 5: 4?

Options

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

Answer

To solve the problem, we need to determine the ratio in which the mixtures from vessels A and B should be taken to achieve a final mixture with a milk to water ratio of 5:4.

### 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 4:5. This means:
- Milk in A = 4 parts
- Water in A = 5 parts
- In vessel B, the ratio of milk to water is 5:1. This means:
- Milk in B = 5 parts
- Water in B = 1 part

2. **Calculate the Fraction of Milk in Each Mixture:**
- For vessel A:
- Total parts in A = 4 + 5 = 9
- Fraction of milk in A = \( \frac{4}{9} \)
- For vessel B:
- Total parts in B = 5 + 1 = 6
- Fraction of milk in B = \( \frac{5}{6} \)

3. **Determine the Desired Ratio of Milk to Water:**
- The desired ratio of milk to water in the final mixture is 5:4.
- Total parts in the desired mixture = 5 + 4 = 9
- Fraction of milk in the desired mixture = \( \frac{5}{9} \)

4. **Set Up the Allegation Method:**
- We will use the allegation method to find the required ratio of the two mixtures.
- Write down the fractions:
- Milk fraction from A = \( \frac{4}{9} \)
- Milk fraction from B = \( \frac{5}{6} \)
- Milk fraction in desired mixture = \( \frac{5}{9} \)

5. **Calculate the Differences:**
- Difference between the fraction from B and the desired fraction:
\[ \frac{5}{6} - \frac{5}{9} = \frac{15}{18} - \frac{10}{18} = \frac{5}{18} \]
- Difference between the fraction from A and the desired fraction:
\[ \frac{5}{9} - \frac{4}{9} = \frac{1}{9} \]

6. **Form the Ratio Using the Differences:**
- The ratio of the quantities taken from A and B is given by the inverse of the differences calculated:
- Ratio = Difference from B : Difference from A
- Thus, the ratio = \( \frac{5}{18} : \frac{1}{9} \)
- To simplify, we can multiply both sides by 18:
- This gives us \( 5 : 2 \)

### Final Answer:
The quantities of mixtures from vessels A and B should be taken in the ratio of **5:2**.