A vessel initially contains 2.5 litres of water and 10 litres of milk. Then, 20\% of the total contents are removed from the vessel. To the remaining mixture, *x* litres of water is added such that the ratio of water to milk reverses. After that, *y* litres of milk is added to again reverse the ratio of water to milk.

Compare the following quantities:

**Quantity I:** Value of *y*

**Quantity II:** Value of *x*

Which of the following is true?

Question

A vessel initially contains 2.5 litres of water and 10 litres of milk. Then, 20\% of the total contents are removed from the vessel. To the remaining mixture, *x* litres of water is added such that the ratio of water to milk reverses. After that, *y* litres of milk is added to again reverse the ratio of water to milk.

Compare the following quantities:

**Quantity I:** Value of *y*

**Quantity II:** Value of *x*

Which of the following is true?

Accepted Answer

**Solution:**

1. Initial quantities:
   - Water = 2.5 litres
   - Milk = 10 litres
   - Total = 12.5 litres

2. After removing 20\% of the contents:
   - Remaining quantity = 80\% of 12.5 = $ \frac{80}{100} \times 12.5 = 10 $ litres

3. The ratio of water to milk remains the same after removal because the mixture is uniform:
   - Water = $ \frac{2.5}{12.5} \times 10 = 2 $ litres
   - Milk = $ \frac{10}{12.5} \times 10 = 8 $ litres

4. After adding *x* litres of water, the ratio reverses. Initially, water : milk = 2 : 8 = 1 : 4.
   - After adding *x* litres water, water = $ 2 + x $, milk = 8
   - The ratio reverses to 4 : 1, so:
     $$
     \frac{2 + x}{8} = \frac{4}{1} \implies 2 + x = 32 \implies x = 30
     $$

5. Next, *y* litres of milk is added to reverse the ratio again.
   - Water remains $ 2 + x = 32 $ litres
   - Milk becomes $ 8 + y $ litres
   - The ratio reverses back to 1 : 4, so:
     $$
     \frac{32}{8 + y} = \frac{1}{4} \implies 32 \times 4 = 8 + y \implies 128 = 8 + y \implies y = 120
     $$

6. Comparing *y* and *x*:
   - *y* = 120 litres
   - *x* = 30 litres

Clearly, $ y > x $.

**Answer:** Quantity I $ \geq $ Quantity II

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