A vessel was containing 80 litres of pure milk. 16 litres of pure milk was taken out and replaced with equal amount of water. 16 litres (of newly formed mixture of water and milk was taken out and then 24 litres of water was added to the mixture. What is the respective ratio between the quantity of milk and water in the final mixture ?

Options

A. `34:21`
B. `28:23`
C. `32:21`
D. `32:23`

Answer

To solve the problem step by step, we will follow the sequence of operations described in the question.

### Step 1: Initial Setup
We start with a vessel containing 80 liters of pure milk.

### Step 2: Remove 16 liters of milk
From the 80 liters of pure milk, we take out 16 liters.
- Remaining milk = 80 liters - 16 liters = 64 liters
- We replace the 16 liters of milk taken out with 16 liters of water.

### Step 3: Composition of the mixture
Now, the vessel contains:
- Milk = 64 liters
- Water = 16 liters
- Total mixture = 64 liters (milk) + 16 liters (water) = 80 liters

### Step 4: Taking out 16 liters of the mixture
Next, we take out 16 liters of the mixture. The ratio of milk to water in the mixture is:
- Milk = 64 liters
- Water = 16 liters
- Ratio of milk to water = 64:16 = 4:1

When we take out 16 liters of the mixture, we need to find out how much milk and how much water is in those 16 liters.

### Step 5: Calculate the amount of milk and water taken out
The total parts in the mixture = 4 (milk) + 1 (water) = 5 parts.
- Each part = 80 liters / 5 parts = 16 liters per part.

Now, in 16 liters of the mixture:
- Milk taken out = 4 parts out of 5 = (4/5) * 16 liters = 12.8 liters
- Water taken out = 1 part out of 5 = (1/5) * 16 liters = 3.2 liters

### Step 6: Remaining quantities after taking out the mixture
After removing 16 liters of the mixture:
- Remaining milk = 64 liters - 12.8 liters = 51.2 liters
- Remaining water = 16 liters - 3.2 liters = 12.8 liters

### Step 7: Adding 24 liters of water
Now, we add 24 liters of water to the remaining mixture:
- New amount of water = 12.8 liters + 24 liters = 36.8 liters

### Step 8: Final quantities
At this point, we have:
- Milk = 51.2 liters
- Water = 36.8 liters

### Step 9: Finding the ratio of milk to water
Now, we need to find the ratio of milk to water:
- Ratio = Milk : Water = 51.2 : 36.8

To simplify this ratio, we can divide both sides by 4.8:
- Milk = 51.2 / 4.8 = 10.6667 (approximately 10.67)
- Water = 36.8 / 4.8 = 7.6667 (approximately 7.67)

To express this in whole numbers, we can multiply both sides by 3:
- Milk = 32
- Water = 23

### Final Ratio
Thus, the final ratio of milk to water is:
- Milk : Water = 32 : 23

### Conclusion
The respective ratio between the quantity of milk and water in the final mixture is **32:23**.

---