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.