A boatman can row 96 km downstream in 8 hours. If the speed of the current is 4 km/h, then in how much time (in hours) will he cover 8 km upstream?

Introduction / Context:
In this problem, we are given the time and distance for a downstream journey and the speed of the current. From this, we must first determine the speed of the boat in still water, and then use it to find the time needed to travel a shorter distance upstream. This exercise reinforces the relationships between downstream speed, upstream speed, boat speed, and stream speed.

Given Data / Assumptions:

• Downstream distance = 96 km.
• Downstream time = 8 hours.
• Speed of the current = 4 km/h.
• We need time to travel 8 km upstream.
• Speeds are uniform, and the river flow is steady.

Concept / Approach:
First we compute the downstream speed from distance and time. Then we use the relation downstream speed = boat speed + current speed to find the boat speed in still water. Once we know the boat speed, upstream speed is boat speed minus current speed. Finally, time upstream is distance divided by upstream speed.

Step-by-Step Solution:
Downstream speed = distance / time = 96 / 8 = 12 km/h. Let b be boat speed in still water and c be current speed. Given c = 4 km/h and b + c = 12 km/h. So b = 12 - 4 = 8 km/h. Upstream speed = b - c = 8 - 4 = 4 km/h. Upstream distance to be covered = 8 km. Time upstream = distance / speed = 8 / 4 = 2 hours.

Verification / Alternative check:
We can verify these speeds with the original downstream journey. With b = 8 km/h and c = 4 km/h, downstream speed is 12 km/h. Time to cover 96 km is 96 / 12 = 8 hours, which matches the given time. For the upstream leg, going at 4 km/h over 8 km clearly takes 2 hours. The calculations are consistent with all the given information.

Why Other Options Are Wrong:
A time of 1 hour would correspond to an upstream speed of 8 km/h, which contradicts the calculated upstream speed of 4 km/h. Times like 1.5 hours or 2.5 hours would also imply inconsistent speeds or contradict the relationship between boat speed, current speed, and downstream speed. Only 2 hours matches the correct upstream speed and the target distance.

Common Pitfalls:
Sometimes learners directly average the known downstream speed and current speed without properly solving for the boat speed. Others mistakenly subtract in the wrong direction when forming upstream and downstream speeds. To avoid such errors, always use the formulas downstream = b + c and upstream = b - c and then carefully substitute values and compute the required time.

Final Answer:
The boatman will take 2 hours to cover 8 km upstream.