The ratio of milk and water in a mixture is `2:1`. How much part of the mixture should be replaced by water so that ratio of milk and water is `5 : 3`?

Options

A. `(1)/(14)`
B. `(1)/(6)`
C. `(1)/(8)`
D. `(1)/(16)`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript and derive the solution systematically.

### Step-by-Step Solution:

1. **Understand the Initial Ratio**:
The initial ratio of milk to water is given as \(2:1\). This means for every 2 parts of milk, there is 1 part of water.

2. **Define the Amounts**:
Let's assume we have a total of 300 liters of the mixture (200 liters of milk and 100 liters of water).
- Milk = 200 liters
- Water = 100 liters

3. **Let X be the Part to be Replaced**:
We need to find out how much part \(X\) of the mixture should be replaced with water to achieve a new ratio of \(5:3\).

4. **Calculate the Amount of Milk and Water Removed**:
When \(X\) liters of the mixture is removed:
- The amount of milk removed = \(\frac{2}{3}X\) (since milk is \(\frac{2}{3}\) of the mixture)
- The amount of water removed = \(\frac{1}{3}X\) (since water is \(\frac{1}{3}\) of the mixture)

5. **Calculate Remaining Milk and Water**:
After removing \(X\) liters:
- Remaining milk = \(200 - \frac{2}{3}X\)
- Remaining water = \(100 - \frac{1}{3}X\)

6. **Add Water**:
We then add \(X\) liters of water back into the mixture:
- New amount of water = \(100 - \frac{1}{3}X + X = 100 + \frac{2}{3}X\)

7. **Set Up the New Ratio**:
We want the new ratio of milk to water to be \(5:3\):
\[ \frac{200 - \frac{2}{3}X}{100 + \frac{2}{3}X} = \frac{5}{3} \]

8. **Cross Multiply**:
Cross multiplying gives us:
\[ 3(200 - \frac{2}{3}X) = 5(100 + \frac{2}{3}X) \]

9. **Expand and Simplify**:
Expanding both sides results in:
\[ 600 - 2X = 500 + \frac{10}{3}X \]
To eliminate the fraction, multiply the entire equation by 3:
\[ 1800 - 6X = 1500 + 10X \]

10. **Combine Like Terms**:
Rearranging gives:
\[ 1800 - 1500 = 10X + 6X \]
\[ 300 = 16X \]

11. **Solve for X**:
\[ X = \frac{300}{16} = \frac{75}{4} = 18.75 \]

12. **Calculate the Fraction of the Mixture**:
The total mixture is 300 liters, so the fraction of the mixture replaced is:
\[ \frac{X}{300} = \frac{75/4}{300} = \frac{75}{1200} = \frac{1}{16} \]

### Final Answer:
The part of the mixture that should be replaced by water is \(\frac{1}{16}\).