A boat takes 2 hours to travel from point A to point B in still water. To determine its speed upstream, which of the following pieces of information are sufficient? A: Distance between A and B. B: Time taken to travel downstream from B to A. C: Speed of the stream of water.

Introduction / Context:
This is a conceptual data sufficiency style question about boats and streams. We know how long the boat takes to travel from A to B in still water, and we are given three possible additional pieces of information. The task is not to calculate a numerical value, but to decide which combinations of information are sufficient to determine the upstream speed of the boat. Understanding how each piece of data relates to the unknown speeds is the key.

Given Data / Assumptions:

• The boat takes 2 hours to go from A to B in still water.
• Let the distance between A and B be d km.
• Let the speed of the boat in still water be b km/h.
• Let the speed of the stream be c km/h.
• Upstream speed = b - c; downstream speed = b + c.
• Information A: the distance d between A and B.
• Information B: the time taken to travel downstream from B to A.
• Information C: the speed c of the stream of water.
• Question: Which combinations of A, B, C are sufficient to find b - c?

Concept / Approach:
From the still water travel, we know the boat covers distance d in 2 hours at speed b, so d = 2b. To get the upstream speed b - c we need enough information to calculate b and c or to deduce b - c directly. We analyse each pair of data: A with B, B with C, and C with A. If any such pair allows us to form equations that lead to a single value for b - c, that pair is sufficient. If the same holds for all three pairs, then any one pair is enough.

Step-by-Step Solution:
Step 1: Use the still water information. From travelling from A to B in still water: distance d = b * 2, so d = 2b. This relationship holds in all cases. Step 2: Consider A and B together. With A, we know d exactly. With B, we know the downstream time t_down. Downstream speed = d / t_down = b + c. Using d = 2b, we get b + c = 2b / t_down. Thus c = 2b / t_down - b. Upstream speed b - c = b - (2b / t_down - b) = 2b - 2b / t_down = 2b(1 - 1 / t_down). Since d and t_down are known and d = 2b, we can solve for b and hence compute b - c. So A and B together are sufficient. Step 3: Consider B and C together. With B, we know t_down, the downstream time. With C, we know c, the stream speed. Downstream speed b + c = d / t_down. From still water travel, d = 2b. So 2b / t_down = b + c. Rearrange: 2b = t_down(b + c) = t_down b + t_down c. Thus b(2 - t_down) = t_down c, giving b = (t_down c) / (2 - t_down). Knowing b and c now allows us to find upstream speed b - c. So B and C together are sufficient. Step 4: Consider C and A together. With A, we know d, the distance. From still water travel, b = d / 2. With C, we know c directly. Upstream speed b - c = d / 2 - c, which is now completely determined. Thus A and C together are also sufficient.
Verification / Alternative check:
We see that with each pair (A,B), (B,C) or (C,A), we end up with enough equations to uniquely determine b - c, the upstream speed. No single item A, B or C on its own is enough, but any pair gives sufficient information.
Why Other Options Are Wrong:
Option a states that only A and B suffice, but we have shown B and C or C and A also suffice. Option b restricts sufficiency to B and C only, which is too narrow. Option c says all are required, which is incorrect because any one pair already gives enough information. Option e limits sufficiency to A and C only, again ignoring the fact that the other pairs work.
Common Pitfalls:
A common mistake is to think that distance must always be known numerically to find speed, ignoring that time and stream speed can also determine the boat speed when combined. Another pitfall is to focus on finding b and c separately, rather than realising that any method that yields b - c directly is also valid and sufficient.
Final Answer:
Any one pair among A and B, B and C, or C and A is sufficient to determine the upstream speed, so the correct choice is Any one pair of A and B, B and C or C and A is sufficient.