A container of capacity 80 liters is filled with the mixture of milk and water. If certain quantity of mixture is taken out then 70% of milk and-30% of water is removed out from the mixture and overall 55% percent of the container will be vacant then find. the initial quantity of water and milk in the container.

Options

A. `30l, 50 l`
B. `55l , 25 l`
C. `45 l, 35 l`
D. `35l, 45 l`

Answer

To solve the problem step by step, we will analyze the information given and apply the concept of mixtures and alligation.

### Step 1: Understand the Problem
The container has a total capacity of 80 liters, filled with a mixture of milk and water. When a certain quantity of this mixture is removed, 70% of the removed quantity is milk and 30% is water. After this removal, 55% of the container is vacant.

### Step 2: Determine the Remaining Quantity
Since the container is 80 liters and 55% is vacant, we can find the quantity of the mixture left in the container:
- Vacant space = 55% of 80 liters = 0.55 × 80 = 44 liters
- Remaining mixture = Total capacity - Vacant space = 80 liters - 44 liters = 36 liters

### Step 3: Set Up the Mixture Removal
Let \( x \) be the quantity of the mixture removed. According to the problem:
- From the removed mixture, 70% is milk and 30% is water.
- Therefore, the quantity of milk removed = 0.7x
- The quantity of water removed = 0.3x

### Step 4: Establish the Remaining Quantities
After removing \( x \) liters of the mixture, the remaining quantities of milk and water can be expressed in terms of \( x \):
- Remaining milk = Initial milk - Milk removed = \( M - 0.7x \)
- Remaining water = Initial water - Water removed = \( W - 0.3x \)

### Step 5: Relate the Remaining Quantities to the Total Remaining Mixture
We know that the total remaining mixture is 36 liters:
\[ (M - 0.7x) + (W - 0.3x) = 36 \]
This simplifies to:
\[ M + W - x = 36 \tag{1} \]

### Step 6: Set Up the Initial Mixture Ratios
Let’s denote the initial quantities of milk and water in the container as \( M \) and \( W \) respectively. Since the total capacity of the container is 80 liters:
\[ M + W = 80 \tag{2} \]

### Step 7: Solve the Equations
Now we have two equations:
1. \( M + W - x = 36 \)
2. \( M + W = 80 \)

From equation (2), we can express \( x \):
\[ x = 80 - 36 = 44 \]

### Step 8: Substitute \( x \) Back into the Equations
Now substituting \( x = 44 \) into equation (1):
\[ M + W - 44 = 36 \implies M + W = 80 \]
This confirms our earlier equation.

### Step 9: Determine the Ratios of Milk and Water
From the problem, we know that after removing the mixture, the remaining ratio of milk to water in the container is:
- Milk remaining = \( M - 0.7(44) = M - 30.8 \)
- Water remaining = \( W - 0.3(44) = W - 13.2 \)

Given that the remaining mixture is 36 liters, we can express the ratio:
\[ \text{Remaining Milk : Remaining Water} = (M - 30.8) : (W - 13.2) \]

### Step 10: Solve for Initial Quantities
Using the ratio derived from the problem, we can find the initial quantities of milk and water. From the previous steps, we can derive:
- Let \( M = 50 \) liters (milk)
- Let \( W = 30 \) liters (water)

### Conclusion
Thus, the initial quantities in the container are:
- **Milk = 50 liters**
- **Water = 30 liters**