Two containers each initially hold 1 litre: the first has pure water and the second has pure milk. Five cups of water are taken from the first container and mixed thoroughly into the second. Then five cups of this new mixture are taken from the second container and poured back into the first. Let A be the proportion of milk in the first container at the end, and B be the proportion of water in the second container at the end. What is the relationship between A and B?

Introduction / Context:
This is a classic puzzle about exchanging equal volumes of liquids between two containers. One container starts with pure water and the other with pure milk. After transferring equal volumes in both directions, we are asked to compare the final proportion of milk in one container with the final proportion of water in the other. The key concept is conservation: what leaves one container in terms of a substance must appear in the other container.

Given Data / Assumptions:
• First container initially: 1 litre of pure water, 0 milk.
• Second container initially: 1 litre of pure milk, 0 water.
• First step: take 5 cups of water from the first container and add to the second, then mix thoroughly.
• Second step: take 5 cups of this mixture from the second container and add back to the first container.
• Cup size is the same in both transfers and is smaller than 1 litre so operations are feasible.

Concept / Approach:
We do not need the exact cup volume; the answer depends only on symmetry and conservation. After the first transfer, only water has moved from the first to the second. After the second transfer, some of this water is moved back along with some milk. The net effect is that the amount of milk present in the first container at the end is equal to the amount of water that remains in the second container at the end. From this, we infer that the proportion of milk in one equals the proportion of water in the other.

Step-by-Step Solution:
Step 1: Initially, first container has 1 litre water and 0 milk; second container has 0 water and 1 litre milk.Step 2: After transferring 5 cups of water from the first to the second, the first container has less water and still no milk. The second now has 1 litre milk plus 5 cups water.Step 3: Mix the second container thoroughly so its composition is uniform.Step 4: When 5 cups of mixture are transferred back to the first container, this mixture contains both milk and water in the same proportions as in the second container at that moment.Step 5: Whatever volume of milk leaves the second container ends up in the first, and the same volume of water going back to the first container reduces the original water imbalance.Step 6: By conservation, the amount of milk that leaves the second and moves to the first equals the amount of water that remains in the second after the exchange, leading to equal fractions.

Verification / Alternative check:
A more formal way is to note that the total amount of water in the system stays constant, as does the total amount of milk. The first container starts with only water, and the second starts with only milk. After the two transfers, the amount of milk in the first container must be exactly equal to the amount of water that has ended up in the second container. Since both containers hold the same total volume (1 litre each) throughout, the fraction of milk in the first equals the fraction of water in the second. Hence A and B are equal.

Why Other Options Are Wrong:
Option A < B or A > B would imply that the net exchanges favored one liquid over the other, which contradicts the symmetry and conservation of volume in both directions. Option cannot be determined is incorrect because, despite not knowing the exact cup size, the qualitative relationship can be deduced from conservation principles.

Common Pitfalls:
Students often think that because water is transferred first, there will always be more water in the second container or more milk in the second container. Others assume we need the exact cup size to solve the problem. In fact, equal volume exchanges and conservation of total quantities make the comparison independent of cup volume, and symmetry gives the relationship directly.

Final Answer:
The proportion of milk in the first container equals the proportion of water in the second container, so A = B.