A man can row three-fourths of a kilometre 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:
This problem focuses on finding the ratio of the rowing speed in still water to the speed of the current using upstream and downstream times for the same distance. It tests your ability to convert minutes to hours, compute speeds, and then combine them using standard boats and streams formulas.

Given Data / Assumptions:
- Distance each way = three fourth of a kilometre = 3/4 km = 0.75 km. - Time taken upstream = 15 minutes. - Time taken downstream = 10 minutes. - Let b be speed in still water (km/h), c speed of current (km/h). - Upstream speed = b - c, downstream speed = b + c.

Concept / Approach:
Speed equals distance divided by time, so we first compute the upstream and downstream speeds from the 0.75 km journeys. Then we use the relations b - c = upstream speed and b + c = downstream speed. Solving these equations gives both b and c, from which the ratio b : c can be simplified to a whole number ratio.

Step-by-Step Solution:
Step 1: Convert times to hours. Upstream: 15 minutes = 15 / 60 = 0.25 hour. Step 2: Upstream speed = distance / time = 0.75 / 0.25 = 3 km/h. Step 3: Downstream: 10 minutes = 10 / 60 hour = 1/6 hour. Step 4: Downstream speed = 0.75 / (1/6) = 0.75 * 6 = 4.5 km/h. Step 5: So b - c = 3 and b + c = 4.5. Step 6: Add equations: (b - c) + (b + c) = 3 + 4.5 gives 2b = 7.5. Step 7: Therefore b = 7.5 / 2 = 3.75 km/h. Step 8: Substitute into b + c = 4.5: 3.75 + c = 4.5, so c = 0.75 km/h. Step 9: Ratio b : c = 3.75 : 0.75. Step 10: Divide both by 0.75 to get 5 : 1.

Verification / Alternative check:
Check with the found speeds. Upstream speed = b - c = 3.75 - 0.75 = 3 km/h. Time for 0.75 km upstream = 0.75 / 3 = 0.25 hour = 15 minutes. Downstream speed = 3.75 + 0.75 = 4.5 km/h, time = 0.75 / 4.5 = 1/6 hour = 10 minutes. Both match the given data, so the ratio 5 : 1 is correct.

Why Other Options Are Wrong:
- Ratios 3 : 1, 4 : 1, 2 : 1, and 6 : 1 do not produce upstream and downstream speeds that match 3 km/h and 4.5 km/h for the given distance and times. - Any incorrect ratio would change either the upstream or downstream time away from 15 and 10 minutes.

Common Pitfalls:
Some learners forget to convert minutes into hours, leading to incorrect speeds. Others may try to directly ratio the times rather than deriving speeds. Always use distance / time to find correct upstream and downstream speeds before forming equations and ratios.

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