Two vessels A and B of equal capacities contain mixtures of milk and water in the ratio `4 : 1` and `3 : 1`, respectively. 25% of the mixture from A is taken out and added to B. After mixing it throughly, an equal amount is taken out from B and added back to A. The ratio of milk to water in vessel A after the second operation is:

Options

A. `79 : 21`
B. `83: 17`
C. `77:23`
D. `81: 19`

Answer

To solve the problem step-by-step, let's break down the operations performed on the mixtures in vessels A and B.

### Step 1: Determine the initial quantities of milk and water in vessels A and B.
- **Vessel A** has a milk to water ratio of 4:1.
- Total volume = 100 liters (assumed).
- Milk in A = \( \frac{4}{4+1} \times 100 = 80 \) liters.
- Water in A = \( \frac{1}{4+1} \times 100 = 20 \) liters.

- **Vessel B** has a milk to water ratio of 3:1.
- Total volume = 100 liters (assumed).
- Milk in B = \( \frac{3}{3+1} \times 100 = 75 \) liters.
- Water in B = \( \frac{1}{3+1} \times 100 = 25 \) liters.

### Step 2: Calculate the amount taken out from vessel A.
- 25% of the mixture from A is taken out.
- Amount taken out = \( 25\% \times 100 = 25 \) liters.

### Step 3: Determine the composition of the 25 liters taken from vessel A.
- The ratio of milk to water in A is 4:1.
- Total parts = 4 + 1 = 5 parts.
- Milk in 25 liters = \( \frac{4}{5} \times 25 = 20 \) liters.
- Water in 25 liters = \( \frac{1}{5} \times 25 = 5 \) liters.

### Step 4: Add the taken mixture to vessel B.
- New quantities in vessel B:
- Milk in B after addition = \( 75 + 20 = 95 \) liters.
- Water in B after addition = \( 25 + 5 = 30 \) liters.

### Step 5: Calculate the new ratio of milk to water in vessel B.
- Total in B = \( 95 + 30 = 125 \) liters.
- Ratio of milk to water in B = \( 95:30 \).
- Simplifying this ratio:
- Divide both by 5: \( \frac{95}{5} : \frac{30}{5} = 19:6 \).

### Step 6: Take out an equal amount from vessel B and add it back to vessel A.
- Amount taken out from B = 25 liters (same as taken out from A).
- Composition of the 25 liters taken from B:
- Total parts = 19 + 6 = 25 parts.
- Milk in 25 liters = \( \frac{19}{25} \times 25 = 19 \) liters.
- Water in 25 liters = \( \frac{6}{25} \times 25 = 6 \) liters.

### Step 7: Update the quantities in vessel A after adding from B.
- New quantities in vessel A:
- Milk in A after addition = \( 80 - 20 + 19 = 79 \) liters.
- Water in A after addition = \( 20 - 5 + 6 = 21 \) liters.

### Step 8: Calculate the final ratio of milk to water in vessel A.
- Final ratio of milk to water in A = \( 79:21 \).

### Final Answer:
The ratio of milk to water in vessel A after the second operation is **79:21**.

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