The ratio of the speed of a boat downstream to the speed of the stream is 9 : 1. If the speed of the current is 3 km/h, how far (in km) will the boat travel upstream in 5 hours?

Introduction / Context:
This problem gives a ratio between the downstream speed of a boat and the speed of the stream. We also know the numerical value of the stream speed and the time the boat travels upstream. The question asks for the upstream distance covered in that time. To solve it, we translate the ratio into actual speeds, determine the boat speed in still water, find the upstream speed, and then use distance = speed * time.

Given Data / Assumptions:

• Ratio of downstream speed to stream speed = 9 : 1.
• Stream speed c = 3 km/h.
• Let downstream speed be d_s km/h and boat speed in still water be b km/h.
• d_s : c = 9 : 1.
• Upstream speed = b - c.
• Time travelled upstream = 5 hours.
• We must find upstream distance in km.

Concept / Approach:
From the ratio d_s : c = 9 : 1 and the known stream speed, we find the downstream speed. Downstream speed is b + c, so we can compute b. Then, using b and c, we obtain the upstream speed b - c. Finally, the upstream distance covered in 5 hours is simply upstream speed multiplied by time. The steps are algebraic but straightforward once the ratio is interpreted correctly.

Step-by-Step Solution:
Step 1: Use the ratio to find downstream speed. d_s : c = 9 : 1 and c = 3 km/h. So d_s = 9 * c = 9 * 3 = 27 km/h. Step 2: Express downstream speed in terms of boat and stream speeds. Downstream speed = b + c = 27. So b + 3 = 27, giving b = 24 km/h. Step 3: Compute upstream speed. Upstream speed = b - c = 24 - 3 = 21 km/h. Step 4: Compute distance covered upstream in 5 hours. Distance = speed * time = 21 * 5 = 105 km.
Verification / Alternative check:
Check the ratio condition. Downstream speed with b = 24 and c = 3 is 24 + 3 = 27 km/h. Ratio of downstream speed to stream speed = 27 : 3 = 9 : 1, which matches the given ratio. Thus the calculated speeds and the resulting distance are consistent.
Why Other Options Are Wrong:
Distances like 100, 98, 109 or 90 km would correspond to upstream speeds that are different from 21 km/h when divided by 5 hours. Those speeds in turn would not match the given downstream to stream speed ratio of 9 : 1.
Common Pitfalls:
A common error is to misinterpret the ratio as boat speed in still water to stream speed, rather than downstream speed to stream speed. Another pitfall is to forget to subtract the stream speed to get the upstream speed, leading to using the downstream speed directly in the distance calculation.
Final Answer:
The boat will travel 105 km upstream in 5 hours.