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, what distance (in km) will the boat travel upstream in 5 hours?

Introduction / Context:
This question combines a ratio relationship involving the downstream speed and the current speed with a simple distance = speed * time calculation for upstream travel. Once the actual speeds are determined, computing the upstream distance over a given time is straightforward.

Given Data / Assumptions:

• Ratio of downstream boat speed to stream speed = 9 : 1.
• Speed of the current (stream) = 3 km/h.
• Let b be the speed of the boat in still water.
• Downstream speed = b + s, where s = 3 km/h.
• Upstream speed = b - s.
• We need the distance travelled upstream in 5 hours.

Concept / Approach:
First, interpret the ratio correctly: downstream speed : stream speed = 9 : 1. Using the actual stream speed, we find the downstream speed and then deduce the boat's still water speed. With the still water speed known, we easily calculate upstream speed and from that compute distance travelled in 5 hours using distance = speed * time.

Step-by-Step Solution:
Step 1: Let downstream speed = 9k and stream speed = 1k for some constant k. Step 2: Given the stream speed is 3 km/h, we have 1k = 3 ⇒ k = 3. Step 3: Therefore downstream speed = 9k = 9 * 3 = 27 km/h. Step 4: Let b be the speed of the boat in still water and s = 3 km/h. Step 5: Downstream speed = b + s ⇒ b + 3 = 27 ⇒ b = 24 km/h. Step 6: Upstream speed = b - s = 24 - 3 = 21 km/h. Step 7: Distance travelled upstream in 5 hours = upstream speed * time = 21 * 5 = 105 km.

Verification / Alternative check:
Check the ratio: downstream speed : stream speed = 27 : 3 = 9 : 1, as given. Also, with still water speed 24 km/h and stream speed 3 km/h, upstream speed 21 km/h is reasonable. Multiplying 21 by 5 gives 105 km, which is consistent and within a realistic range for the given speeds and time.

Why Other Options Are Wrong:
Distances like 100 km, 98 km or 90 km would require smaller upstream speeds than 21 km/h or less time. A distance like 109 km would require an upstream speed greater than 21 km/h over 5 hours, which does not come from the derived still water and stream speeds. Only 105 km matches the product of the correctly calculated upstream speed and the given time.

Common Pitfalls:
A common mistake is to treat the ratio as relating still water speed to stream speed instead of downstream speed to stream speed. Another error is to forget that the stream speed is given directly, which allows you to find the constant k. Always decode ratios carefully and check which actual speeds they refer to before performing final distance calculations.

Final Answer:
The boat will travel 105 km upstream in 5 hours.