A container contains milk and water in the ratio of 4:1. What part of mixture should be taken out and replaced with water such that the container contains milk and water in the ratio of 3 :2?

Options

A. `3/9`
B. `1/4`
C. `2/3`
D. `1/5`

Answer

To solve the problem, we need to determine what part of the mixture should be taken out and replaced with water so that the ratio of milk to water changes from 4:1 to 3:2.

### Step-by-Step Solution:

1. **Initial Setup**:
- Let the initial quantities of milk and water be in the ratio of 4:1.
- Assume the total volume of the mixture is 500 liters (this makes calculations easier).
- Therefore, the amount of milk = \( \frac{4}{5} \times 500 = 400 \) liters.
- The amount of water = \( \frac{1}{5} \times 500 = 100 \) liters.

2. **Let X be the amount of mixture taken out**:
- When we take out X liters of the mixture, we also take out some milk and some water.
- The proportion of milk in the mixture is \( \frac{400}{500} = \frac{4}{5} \).
- The amount of milk taken out = \( \frac{4}{5} \times X = \frac{4X}{5} \).
- The proportion of water in the mixture is \( \frac{100}{500} = \frac{1}{5} \).
- The amount of water taken out = \( \frac{1}{5} \times X = \frac{X}{5} \).

3. **Remaining quantities after taking out X liters**:
- Remaining milk = \( 400 - \frac{4X}{5} \).
- Remaining water = \( 100 - \frac{X}{5} \).

4. **Adding water**:
- We replace the X liters taken out with X liters of water.
- Thus, the new amount of water = \( 100 - \frac{X}{5} + X = 100 + \frac{4X}{5} \).

5. **Final quantities**:
- Final amount of milk = \( 400 - \frac{4X}{5} \).
- Final amount of water = \( 100 + \frac{4X}{5} \).

6. **Setting up the ratio**:
- We want the final ratio of milk to water to be 3:2.
- This gives us the equation:
\[ \frac{400 - \frac{4X}{5}}{100 + \frac{4X}{5}} = \frac{3}{2} \]

7. **Cross-multiplying**:
- Cross-multiplying gives:
\[ 2(400 - \frac{4X}{5}) = 3(100 + \frac{4X}{5}) \]
- Expanding both sides:
\[ 800 - \frac{8X}{5} = 300 + \frac{12X}{5} \]

8. **Combining like terms**:
- Rearranging gives:
\[ 800 - 300 = \frac{12X}{5} + \frac{8X}{5} \]
\[ 500 = \frac{20X}{5} \]
\[ 500 = 4X \]
\[ X = \frac{500}{4} = 125 \text{ liters} \]

9. **Finding the part of the mixture taken out**:
- The total mixture is 500 liters, and we took out 125 liters.
- The part taken out = \( \frac{125}{500} = \frac{1}{4} \).

### Final Answer:
The part of the mixture that should be taken out is \( \frac{1}{4} \).