A vessel contains a mixture of milk and water in the respective ratio of 3: 1. 32 litres of mix ture was taken out and replaced with the same quantity of milk so that the resultant ratio be tween the quantities of milk and water in the mixture was 4:1 respectively. If 10 litres of mix ture is again taken out from the vessel, what is the resultant quantity of water in the mixture? (in litres)

Options

A. 24
B. 30
C. 20
D. 36

Answer

To solve the problem step by step, we will follow the given information and perform calculations accordingly.

### Step 1: Determine the initial quantities of milk and water.
The initial ratio of milk to water is 3:1. Let's assume the total quantity of the mixture is 4x liters (since 3 + 1 = 4).

- Quantity of milk = \(3x\) liters
- Quantity of water = \(x\) liters

### Step 2: Calculate the initial quantities if the total mixture is 32 liters.
Given that the total mixture is 32 liters, we can set up the equation:
\[ 3x + x = 32 \]
This simplifies to:
\[ 4x = 32 \]
Thus,
\[ x = 8 \]

Now, substituting back to find the quantities:
- Quantity of milk = \(3x = 3 \times 8 = 24\) liters
- Quantity of water = \(x = 8\) liters

### Step 3: Calculate the quantities after removing 32 liters of the mixture.
When 32 liters of the mixture is taken out, we need to find out how much milk and water is removed. The mixture has a ratio of 3:1, so:
- Milk removed = \(\frac{3}{4} \times 32 = 24\) liters
- Water removed = \(\frac{1}{4} \times 32 = 8\) liters

### Step 4: Update the quantities after removal.
After removing 32 liters:
- Remaining milk = \(24 - 24 = 0\) liters
- Remaining water = \(8 - 8 = 0\) liters

### Step 5: Add 32 liters of milk back to the mixture.
Now, we replace the 32 liters removed with 32 liters of milk:
- New quantity of milk = \(0 + 32 = 32\) liters
- Quantity of water remains = \(0\) liters

### Step 6: Set up the new ratio of milk to water.
After adding the milk, the new ratio of milk to water is given as 4:1. Let’s denote the quantity of water as \(y\):
\[ \frac{32}{y} = 4 \implies 32 = 4y \implies y = 8 \]
So, the new quantity of water is \(8\) liters.

### Step 7: Calculate the quantity of water after taking out 10 liters of the mixture.
Now, we need to find out how much water is in the 10 liters of mixture that is taken out. The total mixture now consists of 32 liters of milk and 8 liters of water, making a total of 40 liters.

The ratio of milk to water is:
- Milk = 32 liters
- Water = 8 liters

The fraction of water in the mixture is:
\[ \text{Fraction of water} = \frac{8}{40} = \frac{1}{5} \]
Thus, in 10 liters of the mixture, the quantity of water taken out is:
\[ \text{Water taken out} = 10 \times \frac{1}{5} = 2 \text{ liters} \]

### Step 8: Calculate the resultant quantity of water in the mixture.
Initially, there were 8 liters of water. After taking out 2 liters of water:
\[ \text{Remaining water} = 8 - 2 = 6 \text{ liters} \]

### Final Answer:
The resultant quantity of water in the mixture is **6 liters**.

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