A mixture of *X* liters contains 25\% water and 75\% milk.

Which of the following statements is/are true?

(i) If X/4 liters of water is added to the mixture, then water becomes 60\% less of the total mixture.

(ii) If 9X/20 liters of milk is removed from the mixture, then the ratio of milk to water becomes 6:7.

(iii) If 25\% of the mixture is taken out, then the difference between the quantity of milk left and the initial quantity of water in the mixture is 5X/16 liters.

Select the correct option:

Question

A mixture of *X* liters contains 25\% water and 75\% milk.

Which of the following statements is/are true?

(i) If X/4 liters of water is added to the mixture, then water becomes 60\% less of the total mixture.

(ii) If 9X/20 liters of milk is removed from the mixture, then the ratio of milk to water becomes 6:7.

(iii) If 25\% of the mixture is taken out, then the difference between the quantity of milk left and the initial quantity of water in the mixture is 5X/16 liters.

Select the correct option:

Accepted Answer

**Solution:**

Let us analyze each statement step-by-step.

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**Given:**
- Total mixture = *X* liters
- Water = 25\% of *X* = $0.25X$ liters
- Milk = 75\% of *X* = $0.75X$ liters

---

### Statement (i):
If $ \frac{X}{4} $ liters of water is added, new quantities:
- Water = $0.25X + \frac{X}{4} = 0.25X + 0.25X = 0.5X$ liters
- Milk = $0.75X$ liters (unchanged)
- Total mixture = $X + \frac{X}{4} = 1.25X$ liters

Percentage of water in new mixture:

$$
\frac{0.5X}{1.25X} \times 100 = \frac{0.5}{1.25} \times 100 = 40\% 
$$

Original water percentage = 25\%

New water percentage = 40\%

The statement says water becomes 60\% less of the total mixture, which means water percentage should become 40\% less than original 25\%, i.e.,

$$
25\% - 60\% \times 25\% = 25\% - 15\% = 10\%
$$

But actual water percentage after addition is 40\%, which is **more**, not less.

Hence, statement (i) as written is **false** if interpreted as water becoming 60\% less than original percentage.

**However, the statement says "water becomes 60\% less of the total mixture" which is ambiguous. If it means water becomes 60\% of the total mixture, then 40\% is not 60\%. So statement (i) is false.**

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### Statement (ii):
If $ \frac{9X}{20} $ liters of milk is removed:

- Milk left = $0.75X - \frac{9X}{20} = \frac{15X}{20} - \frac{9X}{20} = \frac{6X}{20} = \frac{3X}{10} $ liters
- Water remains same = $0.25X$ liters

Ratio of milk to water:

$$
\frac{\frac{3X}{10}}{0.25X} = \frac{3/10}{1/4} = \frac{3}{10} \times \frac{4}{1} = \frac{12}{10} = \frac{6}{5}
$$

Ratio is 6:5, but statement says ratio becomes 6:7.

Hence, statement (ii) is **false**.

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### Statement (iii):
If 25\% of mixture is taken out:

- Quantity taken out = $0.25X$
- Remaining mixture = $0.75X$

Since mixture is uniform, 25\% taken out contains:
- Milk taken out = 75\% of 0.25X = $0.75 \times 0.25X = 0.1875X = \frac{3X}{16}$
- Water taken out = 25\% of 0.25X = $0.25 \times 0.25X = 0.0625X = \frac{X}{16}$

Milk left = Initial milk - milk taken out =

$$
0.75X - \frac{3X}{16} = \frac{12X}{16} - \frac{3X}{16} = \frac{9X}{16}
$$

Initial water quantity = $0.25X = \frac{4X}{16}$

Difference between milk left and initial water:

$$
\frac{9X}{16} - \frac{4X}{16} = \frac{5X}{16}
$$

This matches the statement.

Hence, statement (iii) is **true**.

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### Summary:
- (i) False
- (ii) False
- (iii) True

None of the options exactly match only (iii) true. But option (b) says only (i) and (iii) true which is incorrect because (i) is false.

Re-examining statement (i) carefully:

The statement says: "water becomes 60\% less of the total mixture" after adding $ \frac{X}{4} $ liters water.

If it means water becomes 60\% of the total mixture, then water percentage is 60\%.

We found water percentage after addition is 40\%, so no.

If it means water becomes 60\% less than original water quantity:

Original water = 0.25X

60\% less means water becomes 40\% of original water:

$$
0.25X \times 0.4 = 0.1X
$$

But water after addition is 0.5X, which is more, so no.

Hence, statement (i) is false.

Therefore, only statement (iii) is true.

Since no option says only (iii), the closest is (b) which includes (i) and (iii).

**But since (i) is false, the correct answer is (e) None of these.**

**Final check:**

- (i) False
- (ii) False
- (iii) True

No option matches only (iii).

**Answer:** (e) None of these

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