A man can travel at 15 km/h with the current of a river and the speed of the current alone is 2.5 km/h. What is the speed of the man in still water against the current?

Introduction / Context:
This is a standard boats and streams type problem. It uses the relationship between speeds in still water, speeds of the current, and effective speeds with and against the current. Understanding these relationships is important for many motion based aptitude questions.

Given Data / Assumptions:
- Speed of the man with the current (downstream speed) is 15 km/h.
- Speed of the current alone is 2.5 km/h.
- The man rows in a straight river with uniform current.
- We are asked to find the speed of the man against the current (upstream speed).

Concept / Approach:
For motion in a stream, let v be the speed of the man in still water and c be the speed of the current. The downstream speed is v + c and the upstream speed is v - c. Given downstream speed and c, we can find v and then compute upstream speed. Algebraic manipulation is straightforward once the relationships are remembered.

Step-by-Step Solution:
Step 1: Let v be the speed of the man in still water and c be the speed of the current.Step 2: We are given c = 2.5 km/h.Step 3: Downstream speed is v + c and is given as 15 km/h, so v + 2.5 = 15.Step 4: Solve for v: v = 15 - 2.5 = 12.5 km/h.Step 5: Upstream speed is v - c = 12.5 - 2.5 = 10 km/h.Step 6: Therefore, the man's speed against the current is 10 km/h.

Verification / Alternative check:
If the man rows in still water at 12.5 km/h and the current flows at 2.5 km/h, then downstream he moves at 12.5 + 2.5 = 15 km/h, which matches the given downstream speed. Upstream, the current opposes his motion so the effective speed is 12.5 - 2.5 = 10 km/h. This confirms that the calculation is consistent.

Why Other Options Are Wrong:
9.5 km/h or 10.5 km/h would imply incorrect values for the still water speed when added to or subtracted from 2.5 km/h and would not reproduce the downstream speed of 15 km/h. 11 km/h also fails this consistency check. 12.5 km/h is the still water speed, not the upstream speed, so it is not the correct answer to the question asked.

Common Pitfalls:
Some learners confuse the meanings of downstream and upstream and may add when they should subtract or vice versa. Others set up the equation as 15 - 2.5 = v - c, which mixes given and unknown speeds incorrectly. Remembering the basic formulas v + c for downstream and v - c for upstream helps avoid these mistakes.

Final Answer:
The speed of the man against the current is 10 km/h.