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. Here’s a step-by-step solution:

### Step 1: Understand the initial mixture
The initial mixture consists of:
- Water = 6 parts
- Milk = 10 parts

The total parts of the mixture = 6 + 10 = 16 parts.

### Step 2: Determine the current ratio
The current ratio of water to milk is:
- Water : Milk = 6 : 10 = 3 : 5

### Step 3: Define the target ratio
We want the final mixture to be half water and half milk. This means:
- Water : Milk = 1 : 1

### Step 4: Let x be the amount of mixture drawn off
Let x be the amount of the mixture drawn off from the container. When we draw off x parts of the mixture, the amount of water and milk removed will be in the same ratio as the original mixture.

### Step 5: Calculate the amounts removed
From the mixture drawn off (x parts):
- Water removed = (3/8) * x (since 3 parts out of 8 total parts are water)
- Milk removed = (5/8) * x (since 5 parts out of 8 total parts are milk)

### Step 6: Calculate the remaining amounts after drawing off
After removing x parts:
- Remaining water = 6 - (3/8)x
- Remaining milk = 10 - (5/8)x

### Step 7: Add water to the mixture
After drawing off x parts, we replace it with x parts of water. Therefore, the new amount of water becomes:
- New water = Remaining water + x = (6 - (3/8)x) + x = 6 - (3/8)x + (8/8)x = 6 + (5/8)x

### Step 8: Set up the equation for the target ratio
We want the final amounts of water and milk to be equal:
\[ 6 + \frac{5}{8}x = 10 - \frac{5}{8}x \]

### Step 9: Solve for x
Combine like terms:
\[ 6 + \frac{5}{8}x + \frac{5}{8}x = 10 \]
\[ 6 + \frac{10}{8}x = 10 \]
\[ \frac{10}{8}x = 10 - 6 \]
\[ \frac{10}{8}x = 4 \]
\[ x = 4 * \frac{8}{10} \]
\[ x = \frac{32}{10} \]
\[ x = 3.2 \]

### Step 10: Calculate the fraction of the mixture drawn off
The total mixture is 16 parts, so the fraction of the mixture drawn off is:
\[ \text{Fraction} = \frac{x}{\text{Total parts}} = \frac{3.2}{16} = \frac{1}{5} \]

### Final Answer
The amount of the mixture that must be drawn off and replaced with water is \( \frac{1}{5} \) of the total mixture.

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