Two vessels P and Q contain mixtures of milk and water in the ratios 3 : 2 and 5 : 3 respectively. If 10 litres of mixture are removed from vessel P and poured into vessel Q, and then the ratio of milk to water in vessel Q becomes 8 : 5, what was the initial quantity of water in vessel Q?

Introduction / Context:
This question combines ratio concepts with mixture transfer between two vessels. We are given initial ratios of milk and water in both vessels, a quantity of mixture transferred from one vessel to another, and the final ratio in one vessel. We must work backwards to find the original amount of water in that vessel. This tests algebraic modeling of mixtures and ratios.

Given Data / Assumptions:
• Vessel P has milk : water = 3 : 2.
• Vessel Q has milk : water = 5 : 3.
• From vessel P, 10 litres of its mixture is removed and poured into vessel Q.
• After this transfer, the ratio milk : water in vessel Q becomes 8 : 5.
• We must find the initial quantity of water in vessel Q (in litres).

Concept / Approach:
We use the fact that when we remove 10 litres from P with ratio 3 : 2, the removed liquid contains milk and water in the same ratio, i.e., 6 litres milk and 4 litres water. For vessel Q, we let its initial volume be V litres. Using its initial ratio, we can express initial milk and water in Q in terms of V. After adding 6 litres of milk and 4 litres of water from P, we use the final ratio 8 : 5 to set up and solve an equation for V. From V, we compute the initial water quantity in Q.

Step-by-Step Solution:
Step 1: In vessel P, ratio 3 : 2 means that in 10 litres of mixture, milk = (3/5) * 10 = 6 litres and water = (2/5) * 10 = 4 litres.Step 2: Let initial total mixture in vessel Q be V litres. With ratio 5 : 3, milk in Q is (5/8)V and water is (3/8)V.Step 3: After transferring from P to Q, milk in Q = (5/8)V + 6 and water in Q = (3/8)V + 4.Step 4: The final ratio in Q is milk : water = 8 : 5, so ((5/8)V + 6) / ((3/8)V + 4) = 8 / 5.Step 5: Cross-multiply: 5 * ((5/8)V + 6) = 8 * ((3/8)V + 4).Step 6: Simplify left side: 5 * (5V/8 + 6) = (25V/8) + 30. Simplify right side: 8 * (3V/8 + 4) = 3V + 32.Step 7: Set (25V/8) + 30 = 3V + 32. Multiply by 8: 25V + 240 = 24V + 256, so V = 16.Step 8: Initial water in Q = (3/8)V = (3/8)*16 = 6 litres.

Verification / Alternative check:
With V = 16, initial milk in Q = (5/8)*16 = 10 litres, water = 6 litres. After adding 6 litres milk and 4 litres water from P, we get milk = 16 litres and water = 10 litres. The ratio 16 : 10 simplifies to 8 : 5, which matches the final ratio given. Thus, the calculation is consistent and confirms that the initial water in Q is 6 litres.

Why Other Options Are Wrong:
Option 10 litres or 12 litres for initial water would lead to different initial totals V and would not result in a final milk : water ratio of 8 : 5 after adding 6 litres milk and 4 litres water.
Option 16 litres is impossible because it would imply all of vessel Q initially is water, which contradicts the given ratio 5 : 3.

Common Pitfalls:
Common mistakes include assuming that only water or only milk is transferred, forgetting that the transferred 10 litres maintains the ratio from P. Another error is ignoring that V is the total mixture in Q, not directly the water content. Setting up the equation carefully using the correct expressions for milk and water before and after transfer is crucial.

Final Answer:
The initial quantity of water in vessel Q was 6 litres.