There are two containers X and Y. X contains 100 ml of milk and Y contains 100 ml of water. 20 ml of milk from X is transferred to Y. After mixing well, 20 ml of the mixture in Y is transferred back to X. If m denotes the proportion of milk in X and n denotes the proportion of water in Y, then which one of the following is correct?

Here's a step-by-step breakdown:

1. Initial State:

   • Container X: 100 ml Milk, 0 ml Water.
   • Container Y: 0 ml Milk, 100 ml Water.

2. After 1st Transfer (20 ml milk from X to Y):

   • Container X: 80 ml Milk (100 - 20), 0 ml Water.
   • Container Y: 20 ml Milk, 100 ml Water. Total volume in Y is 120 ml.

3. After 2nd Transfer (20 ml of mixture from Y to X):

   • First, determine the composition of the mixture in Y:
     • Proportion of Milk in Y = 20 ml / 120 ml = 1/6.
     • Proportion of Water in Y = 100 ml / 120 ml = 5/6.
   • Now, 20 ml of this mixture is transferred back to X:
     • Amount of Milk transferred back to X = 20 ml * (1/6) = 10/3 ml.
     • Amount of Water transferred back to X = 20 ml * (5/6) = 50/3 ml.

4. Final State of Containers:

   • Container X:
     • Milk = 80 ml (remaining) + 10/3 ml (transferred) = 240/3 + 10/3 = 250/3 ml.
     • Water = 0 ml + 50/3 ml (transferred) = 50/3 ml.
     • Total volume in X = 250/3 + 50/3 = 300/3 = 100 ml.
   • Container Y:
     • Milk = 20 ml (initial in Y) - 10/3 ml (transferred out) = 60/3 - 10/3 = 50/3 ml.
     • Water = 100 ml (initial in Y) - 50/3 ml (transferred out) = 300/3 - 50/3 = 250/3 ml.
     • Total volume in Y = 50/3 + 250/3 = 300/3 = 100 ml.

5. Calculate m and n:

   • m = proportion of milk in X = (Milk in X) / (Total volume in X) = (250/3 ml) / 100 ml = 250/300 = 5/6.
   • n = proportion of water in Y = (Water in Y) / (Total volume in Y) = (250/3 ml) / 100 ml = 250/300 = 5/6.

6. Conclusion:

   • Since m = 5/6 and n = 5/6, then m = n.

Therefore, option A is correct.