The ratio of milk and water in a mixture is `7:5`. How much part of the mixture should be replaced by water so that ratio of milk and water is `2:3`?

Options

A. `(11)/(35)`
B. `(11)/(25)`
C. `(13)/(24)`
D. `(13)/(36)`

Answer

To solve the problem of how much part of the mixture should be replaced by water so that the ratio of milk to water changes from 7:5 to 2:3, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Initial Mixture:**
- Let the total quantity of the mixture be \( 1200 \) parts (for simplicity).
- Since the ratio of milk to water is \( 7:5 \):
- Milk = \( \frac{7}{12} \times 1200 = 700 \) parts
- Water = \( \frac{5}{12} \times 1200 = 500 \) parts

2. **Let \( x \) be the part of the mixture replaced by water:**
- When \( x \) parts of the mixture are removed, both milk and water will be removed in the same ratio.
- The amount of milk removed = \( \frac{7}{12}x \)
- The amount of water removed = \( \frac{5}{12}x \)

3. **Calculate the Remaining Milk and Water:**
- Remaining milk = \( 700 - \frac{7}{12}x \)
- Remaining water = \( 500 - \frac{5}{12}x \)

4. **Add \( x \) parts of water:**
- New amount of water = \( 500 - \frac{5}{12}x + x = 500 - \frac{5}{12}x + \frac{12}{12}x = 500 + \frac{7}{12}x \)

5. **Set Up the New Ratio:**
- The new ratio of milk to water should be \( 2:3 \).
- Therefore, we can write the equation:
\[ \frac{700 - \frac{7}{12}x}{500 + \frac{7}{12}x} = \frac{2}{3} \]

6. **Cross Multiply to Solve for \( x \):**
\[ 3 \left( 700 - \frac{7}{12}x \right) = 2 \left( 500 + \frac{7}{12}x \right) \]
- Expanding both sides:
\[ 2100 - \frac{21}{12}x = 1000 + \frac{14}{12}x \]
- Rearranging gives:
\[ 2100 - 1000 = \frac{21}{12}x + \frac{14}{12}x \]
\[ 1100 = \frac{35}{12}x \]

7. **Solve for \( x \):**
\[ x = \frac{1100 \times 12}{35} = \frac{13200}{35} = 377.14 \text{ (approximately)} \]

8. **Determine the Fraction of the Total Mixture:**
- The total mixture is \( 1200 \) parts.
- The fraction of the mixture replaced by water:
\[ \frac{x}{1200} = \frac{377.14}{1200} \approx 0.3143 \]

9. **Convert to a Fraction:**
- To express this as a fraction, we can simplify:
\[ \frac{377.14}{1200} \approx \frac{11}{35} \]

### Final Answer:
The part of the mixture that should be replaced by water is \( \frac{11}{35} \).