A man rows 3/4 km against the current in 15 minutes and returns the same distance with the current in 10 minutes. What is the ratio of his speed in still water to the speed of the current?

Introduction / Context:
Here we are asked to find the ratio of a man’s rowing speed in still water to the speed of the current. The man rows a short distance upstream and downstream with different times for each leg. By using distance over time we find effective speeds and then use the boats and streams relationships to derive the required ratio. Ratios are often tested to avoid computing exact decimal values.

Given Data / Assumptions:

• Distance each way = 3/4 km.
• Upstream time = 15 minutes.
• Downstream time = 10 minutes.
• Speeds are constant in both directions.
• We want the ratio speed in still water : stream speed.

Concept / Approach:
If b is the man’s speed in still water and c is the speed of the current, then:

• Upstream speed = b - c.
• Downstream speed = b + c.

We first compute the numerical values of upstream and downstream speeds from distance and time. Then we solve for b and c and finally express their ratio b : c in simplest form.

Step-by-Step Solution:
Convert times to hours: 15 minutes = 15 / 60 = 1/4 hour, 10 minutes = 10 / 60 = 1/6 hour. Upstream speed = distance / time = (3/4) / (1/4) = 3 km/h. Downstream speed = (3/4) / (1/6) = (3/4) * 6 = 18 / 4 = 4.5 km/h. Let b be speed in still water and c be stream speed. Then b - c = 3 and b + c = 4.5. Add the equations: 2b = 7.5 so b = 7.5 / 2 = 3.75 km/h. Subtract the equations: 2c = 1.5 so c = 1.5 / 2 = 0.75 km/h. Thus the ratio b : c = 3.75 : 0.75. Divide both numbers by 0.75 to obtain 5 : 1.

Verification / Alternative check:
Using b = 3.75 km/h and c = 0.75 km/h, upstream speed is 3.75 - 0.75 = 3 km/h, downstream speed is 3.75 + 0.75 = 4.5 km/h. For a distance of 3/4 km, upstream time is (3/4) / 3 = 1/4 hour or 15 minutes, and downstream time is (3/4) / 4.5 = 1/6 hour or 10 minutes. These match the problem statement, confirming that the speeds and ratio are correct.

Why Other Options Are Wrong:
Ratios 3:1, 4:1, and 2:1 would correspond to different pairs of b and c that do not produce the observed upstream and downstream times. For example, a 3:1 ratio might lead to speeds that yield equal or different time differences. Only the 5:1 ratio reconstructs the correct upstream and downstream times for the given distance.

Common Pitfalls:
One common mistake is to forget to convert minutes to hours, causing incorrect speeds. Another is to try to directly ratio the given times without first computing speeds. Also, some learners might mistakenly compute stream speed as the difference of times instead of the difference of speeds. Clear steps and consistent use of units prevent these errors.

Final Answer:
The ratio of the man’s speed in still water to the speed of the current is 5:1.