A container is filled with liquid, 6 part of which are water 10 part milk. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half milk?

Options

A. `1/3`
B. `1/7`
C. `1/5`
D. `1/8`

Answer

To solve the problem, we need to determine how much of the mixture must be drawn off and replaced with water so that the mixture becomes half water and half milk.

### Step-by-Step Solution:

1. **Understand the Initial Composition**:
- The container has a total of \(6\) parts water and \(10\) parts milk.
- Therefore, the total parts of the mixture = \(6 + 10 = 16\) parts.

2. **Determine the Desired Composition**:
- We want the final mixture to be half water and half milk.
- Since the total mixture is \(16\) parts, half of it should be \(8\) parts water and \(8\) parts milk.

3. **Let \(x\) be the amount of mixture drawn off**:
- When we draw off \(x\) parts of the mixture, we are removing both water and milk in the same ratio as they are present in the mixture.
- The fraction of water in the mixture = \(\frac{6}{16} = \frac{3}{8}\).
- The fraction of milk in the mixture = \(\frac{10}{16} = \frac{5}{8}\).

4. **Calculate the Amount of Water and Milk Removed**:
- Amount of water removed = \(x \times \frac{3}{8} = \frac{3x}{8}\).
- Amount of milk removed = \(x \times \frac{5}{8} = \frac{5x}{8}\).

5. **Calculate the Remaining Amounts After Drawing Off**:
- Remaining water = Initial water - Water removed = \(6 - \frac{3x}{8}\).
- Remaining milk = Initial milk - Milk removed = \(10 - \frac{5x}{8}\).

6. **Replace the Drawn Mixture with Water**:
- After drawing off \(x\) parts, we replace it with \(x\) parts of water.
- New amount of water = Remaining water + \(x\) = \(6 - \frac{3x}{8} + x = 6 + \frac{5x}{8}\).
- The amount of milk remains the same = \(10 - \frac{5x}{8}\).

7. **Set Up the Equation for Final Composition**:
- For the final mixture to be half water and half milk:
- \(6 + \frac{5x}{8} = 10 - \frac{5x}{8}\).

8. **Solve the Equation**:
- Combine like terms:
\[ 6 + \frac{5x}{8} + \frac{5x}{8} = 10 \]
\[ 6 + \frac{10x}{8} = 10 \]
\[ \frac{10x}{8} = 10 - 6 \]
\[ \frac{10x}{8} = 4 \]
\[ 10x = 32 \]
\[ x = \frac{32}{10} = 3.2 \]

9. **Convert to Fraction**:
- Since the options are in fractions, we can express \(3.2\) as a fraction:
\[ x = \frac{32}{10} = \frac{16}{5} \]

10. **Final Answer**:
- The amount of the mixture that must be drawn off and replaced with water is \( \frac{16}{5} \) parts.