A boat travels 4 km upstream and 4 km downstream in a total time of 1 hour. The same boat travels 5 km downstream and 3 km upstream in a total time of 55 minutes. Based on this information, what is the speed of the boat in still water (in km/h)?

Introduction / Context:
This boats and streams problem asks you to determine the speed of a boat in still water when it makes two different trips involving both upstream and downstream motion. Because upstream and downstream speeds are affected by the current, we use the concept of relative speed to set up equations and solve for the unknown speed of the boat in still water.

Given Data / Assumptions:

• The boat goes 4 km upstream and 4 km downstream in a total time of 1 hour.
• It then goes 5 km downstream and 3 km upstream in a total time of 55 minutes (which is 11/12 hour).
• Let the speed of the boat in still water be b km/h.
• Let the speed of the stream be s km/h.
• Upstream speed = (b - s) km/h, downstream speed = (b + s) km/h.

Concept / Approach:
We use the idea that time = distance / speed. For each mixed trip, we express the total time as the sum of times for upstream and downstream segments. This gives us two equations in terms of b and s. Solving these simultaneous equations gives the values of b and s. We are finally interested in b, the speed of the boat in still water.

Step-by-Step Solution:
Let b = speed of boat in still water, s = speed of stream. Upstream speed = b - s, downstream speed = b + s. From the first trip: 4 / (b - s) + 4 / (b + s) = 1. From the second trip: 5 / (b + s) + 3 / (b - s) = 11 / 12. Multiply the first equation by (b - s)(b + s): 4(b + s) + 4(b - s) = (b^2 - s^2). So 4b + 4s + 4b - 4s = b^2 - s^2 ⇒ 8b = b^2 - s^2. (1) For the second equation, multiply by (b - s)(b + s): 5(b - s) + 3(b + s) = (11 / 12)(b^2 - s^2). This gives 5b - 5s + 3b + 3s = (11 / 12)(b^2 - s^2). So 8b - 2s = (11 / 12)(b^2 - s^2). (2) From (1), b^2 - s^2 = 8b. Substitute into (2): 8b - 2s = (11 / 12) * 8b = (88 / 12)b = (22 / 3)b. Thus, 8b - 2s = (22 / 3)b ⇒ multiply by 3: 24b - 6s = 22b ⇒ 2b = 6s ⇒ b = 3s. Substitute b = 3s into (1): (3s)^2 - s^2 = 8b ⇒ 9s^2 - s^2 = 8 * 3s ⇒ 8s^2 = 24s. Since s ≠ 0, divide both sides by 8s: s = 3 km/h and b = 3 * 3 = 9 km/h.

Verification / Alternative check:
With b = 9 and s = 3, upstream speed is 6 km/h, downstream speed is 12 km/h. First trip time = 4/6 + 4/12 = 2/3 + 1/3 = 1 hour, correct. Second trip time = 5/12 + 3/6 = 5/12 + 1/2 = 5/12 + 6/12 = 11/12 hour, which matches 55 minutes. So the solution is consistent.

Why Other Options Are Wrong:
Speeds such as 6.5, 7.75, 8.25 or 10.5 km/h do not satisfy both time equations simultaneously. Substituting any of these values into the expressions for upstream and downstream times leads to totals different from 1 hour and 55 minutes. Only 9 km/h satisfies both conditions exactly.

Common Pitfalls:
Students often try to average distances or speeds without using proper equations, or they forget to convert 55 minutes into hours. Another frequent mistake is mixing upstream and downstream speeds. Always define variables clearly, write the time equations carefully and solve them systematically.

Final Answer:
The speed of the boat in still water is 9 km/h.