A tank contains 384 liters of a mixture of milk and water in the ratio of 15 : 17. If *X* liters of the mixture is removed and then *(X - 14)* liters of milk and *(X - 34)* liters of water are added to the remaining mixture, the ratio of milk to water becomes equal. What percentage of the total final mixture is the quantity of milk in the original mixture?

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Accepted Answer

**Solution:**

1. **Initial quantities:**
   - Total mixture = 384 liters
   - Ratio of milk to water = 15 : 17

2. **Calculate initial quantities of milk and water:**
   - Total parts = 15 + 17 = 32
   - Milk = $ \frac{15}{32} \times 384 = 180 $ liters
   - Water = $ \frac{17}{32} \times 384 = 204 $ liters

3. **After removing *X* liters of mixture:**
   - Milk removed = $ \frac{15}{32} \times X = \frac{15X}{32} $
   - Water removed = $ \frac{17}{32} \times X = \frac{17X}{32} $

4. **Remaining quantities after removal:**
   - Milk left = $ 180 - \frac{15X}{32} $
   - Water left = $ 204 - \frac{17X}{32} $

5. **Adding milk and water:**
   - Milk added = $ X - 14 $
   - Water added = $ X - 34 $

6. **New quantities:**
   - Milk = $ 180 - \frac{15X}{32} + (X - 14) = 180 - \frac{15X}{32} + X - 14 $
   - Water = $ 204 - \frac{17X}{32} + (X - 34) = 204 - \frac{17X}{32} + X - 34 $

7. **Simplify:**
   - Milk = $ 166 + X - \frac{15X}{32} = 166 + \frac{32X}{32} - \frac{15X}{32} = 166 + \frac{17X}{32} $
   - Water = $ 170 + X - \frac{17X}{32} = 170 + \frac{32X}{32} - \frac{17X}{32} = 170 + \frac{15X}{32} $

8. **Given that the ratio of milk to water becomes equal:**
   $$
   166 + \frac{17X}{32} = 170 + \frac{15X}{32}
   $$

9. **Solve for *X*:**
   $$
   166 + \frac{17X}{32} = 170 + \frac{15X}{32} \
   \frac{17X}{32} - \frac{15X}{32} = 170 - 166 \
   \frac{2X}{32} = 4 \
   \frac{X}{16} = 4 \
   X = 64
   $$

10. **Calculate final total mixture:**
    - Removed mixture = 64 liters
    - Added milk = 64 - 14 = 50 liters
    - Added water = 64 - 34 = 30 liters

Final volume = Initial volume - removed + added = 384 - 64 + 50 + 30 = 400 liters

11. **Milk in original mixture = 180 liters**

12. **Find percentage of milk in original mixture relative to final mixture:**
    $$
    \frac{180}{400} \times 100 = 45\% 
    $$

13. **Check options:**
    The closest correct answer is 45\%, but since the ratio becomes equal, the milk and water quantities are equal in the final mixture, so milk is half of 400 liters = 200 liters, which contradicts the original milk quantity.

Re-examining step 11, the question asks: "quantity of milk in original mixture is what percent of total final mixture?"

So, milk in original mixture = 180 liters
    Total final mixture = 400 liters

Percentage = $ \frac{180}{400} \times 100 = 45\% $

**Answer:** 45\%

**Note:** The correct option is (c) 45\%.

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