A man rows downstream from his starting point to a destination and then rows back upstream to the starting point in a total of 5 hours. If the speed of the boat in still water is 10 km/h and the speed of the stream is 4 km/h, what is the distance between the starting point and the destination?

Introduction / Context:
This question describes a round trip on a river: a man travels from his starting point to a destination downstream and then returns upstream. You are given the total time for the complete up-and-down journey, as well as the speed of the boat in still water and the speed of the stream. You must determine the one-way distance to the destination. This is a standard application of relative speed and time equations in boats and streams problems.

Given Data / Assumptions:

• Speed of boat in still water b = 10 km/h.
• Speed of stream s = 4 km/h.
• Downstream speed = b + s = 14 km/h.
• Upstream speed = b − s = 6 km/h.
• Total time for downstream plus upstream journey = 5 hours.
• Let one-way distance to the destination be D km.

Concept / Approach:
We express the total time as the sum of the downstream and upstream travel times for the same distance D:

• Downstream time = D / 14 hours.
• Upstream time = D / 6 hours.

The sum of these two times is given as 5 hours. So we solve the equation D/14 + D/6 = 5 for D.

Step-by-Step Solution:
Step 1: Write the total time equation: D/14 + D/6 = 5.Step 2: Factor D: D(1/14 + 1/6) = 5.Step 3: Compute 1/14 + 1/6 = (3/42 + 7/42) = 10/42 = 5/21.Step 4: So D * (5/21) = 5.Step 5: Divide both sides by (5/21): D = 5 * (21/5) = 21.Step 6: Therefore, the distance between the starting point and the destination is 21 km.

Verification / Alternative check:
Substitute D = 21 back into the times. Downstream time = 21 / 14 = 1.5 hours. Upstream time = 21 / 6 = 3.5 hours. Total time = 1.5 + 3.5 = 5 hours, exactly as given in the problem. This confirms that D = 21 km is correct.

Why Other Options Are Wrong:
If D were 16, 18, 25 or 12 km, the total time D/14 + D/6 would not equal 5 hours. For instance, with D = 16, the total time is less than 5 hours, and with D = 25, it exceeds 5 hours. Only D = 21 satisfies the equation derived from the given speeds and total time.

Common Pitfalls:

• Using 10 km/h and 4 km/h directly as the upstream and downstream speeds, instead of adding or subtracting the stream speed.
• Forgetting that both the downstream and upstream distances are D, not different values.
• Arithmetic mistakes when adding the fractions 1/14 and 1/6.
• Trying to approximate instead of forming and solving the exact equation.

Final Answer:
The destination is 21 km away from the starting point.