There are two vessels A and B containing 25 litres each of pure milk and pure water respectively. 5 litres of milk from A is taken and poured into B, then 6 litres of mixture from B is taken and poured in A. What is the ratio of water in A and B respectively :

Options

A. 0.17013888888889
B. 0.044444444444444
C. 0.21111111111111
D. 0.085416666666667

Answer

To solve the problem step-by-step, we will analyze the operations performed on vessels A and B.

### Step 1: Initial Setup
- Vessel A contains 25 liters of pure milk.
- Vessel B contains 25 liters of pure water.

### Step 2: Transfer 5 Liters of Milk from A to B
- When we take 5 liters of milk from vessel A and pour it into vessel B:
- Vessel A now has \(25 - 5 = 20\) liters of milk.
- Vessel B now has \(25 + 5 = 30\) liters of mixture (25 liters of water + 5 liters of milk).

### Step 3: Composition of Mixture in B
- The mixture in vessel B consists of:
- 5 liters of milk
- 25 liters of water
- The total volume in B is 30 liters (5 liters of milk + 25 liters of water).
- The ratio of milk to water in B is \(5:25\) or \(1:5\).

### Step 4: Calculate the Amount of Milk and Water in 6 Liters Taken from B
- We take 6 liters of this mixture from vessel B.
- The fraction of milk in B is \( \frac{5}{30} = \frac{1}{6} \).
- The fraction of water in B is \( \frac{25}{30} = \frac{5}{6} \).

- Therefore, in 6 liters taken from B:
- Milk = \(6 \times \frac{1}{6} = 1\) liter
- Water = \(6 \times \frac{5}{6} = 5\) liters

### Step 5: Transfer 6 Liters of Mixture from B to A
- After transferring 6 liters from B to A:
- Vessel A now has:
- Milk: \(20 + 1 = 21\) liters
- Water: \(0 + 5 = 5\) liters

### Step 6: Final Composition in Both Vessels
- Vessel A:
- Milk: 21 liters
- Water: 5 liters

- Vessel B:
- Initially, it had 30 liters (5 liters milk + 25 liters water).
- After taking out 6 liters (1 liter milk + 5 liters water), it now has:
- Milk: \(5 - 1 = 4\) liters
- Water: \(25 - 5 = 20\) liters

### Step 7: Final Ratios of Water in A and B
- Water in A = 5 liters
- Water in B = 20 liters

### Step 8: Calculate the Ratio of Water in A and B
- The ratio of water in A to water in B is:
\[ \text{Ratio} = \frac{\text{Water in A}}{\text{Water in B}} = \frac{5}{20} = \frac{1}{4} \]

### Conclusion
The ratio of water in vessels A and B is \(1:4\).

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