A boat running upstream takes 8 hours 48 minutes to cover a certain distance, while it takes 4 hours to cover the same distance running downstream. What is the ratio of the speed of the boat in still water to the speed of the water current, respectively?

Introduction / Context:
This problem deals with the ratio of the boat's speed in still water to the speed of the current, based on the times needed to travel the same distance upstream and downstream. It is a standard type of boats and streams question involving relative speed and ratios.

Given Data / Assumptions:

Let the one-way distance be d km.
Time upstream = 8 hours 48 minutes = 8 + 48/60 = 8.8 hours = 44/5 hours.
Time downstream = 4 hours.
Let boat speed in still water be b km/h.
Let current speed be c km/h.
Upstream speed = b - c; downstream speed = b + c.
Distance d is the same in both directions.

Concept / Approach:
Speed is distance divided by time. We can express upstream and downstream speeds in terms of d and the given times. Then the ratio of upstream to downstream speeds equals the ratio of (b - c) to (b + c). Solving this ratio equation gives b/c, which is the required ratio of the boat's speed in still water to the current speed.

Step-by-Step Solution:
Step 1: Compute upstream and downstream speeds in terms of d. Upstream speed v_u = d / (44/5) = 5d/44. Downstream speed v_d = d / 4. Step 2: Form the ratio of upstream to downstream speeds. v_u / v_d = (5d/44) / (d/4) = (5d/44) * (4/d) = 20/44 = 5/11. So (b - c) / (b + c) = 5/11. Step 3: Solve for the ratio b : c. (b - c) / (b + c) = 5/11 ⇒ 11(b - c) = 5(b + c). 11b - 11c = 5b + 5c ⇒ 11b - 5b = 11c + 5c ⇒ 6b = 16c. Thus b/c = 16/6 = 8/3. Therefore, the required ratio b : c = 8 : 3.

Verification / Alternative check:
Let us assume c = 3 units, then b = 8 units (same units, e.g. km/h). Upstream speed = 8 - 3 = 5 units; downstream speed = 8 + 3 = 11 units. Distance d is proportional to speed × time. Using upstream time 44/5 hours and speed 5 units gives d = 5 * 44/5 = 44 units. Using downstream time 4 hours and speed 11 units gives d = 11 * 4 = 44 units again, confirming consistency.

Why Other Options Are Wrong:
7:4 and 11:4 do not satisfy the equation (b - c)/(b + c) = 5/11 when you plug in those ratios for b and c.
4:7 implies the boat is slower than the current, which would make upstream motion impossible or extremely slow and does not match the given times.

Common Pitfalls:
Errors often occur when converting 8 hours 48 minutes into hours; forgetting that 48 minutes is 4/5 of an hour leads to wrong times and speeds. Another mistake is to invert the ratio, taking downstream to upstream instead of upstream to downstream. Carefully setting up the ratio (b - c)/(b + c) and doing the algebra step by step ensures the correct answer.

Final Answer:
The ratio of the speed of the boat in still water to the speed of the current is 8:3.