A man can row a boat upstream at 16 km/h and downstream at 24 km/h in a river. What is the speed of the current (in km/h)?

Introduction / Context:
This boats and streams question gives the rowing speeds of a man upstream and downstream and asks for the speed of the current. When a boat moves upstream, the current opposes the motion, and when it moves downstream, the current helps the motion. Using the two given effective speeds, we can extract the speed of the boat in still water and the speed of the current using simple algebraic relationships.

Given Data / Assumptions:

• Upstream speed = 16 km/h.
• Downstream speed = 24 km/h.
• Let the speed of the boat in still water be b km/h.
• Let the speed of the current be c km/h.
• Upstream speed = b - c and downstream speed = b + c.
• We must find c, the speed of the current.

Concept / Approach:
In problems where both upstream and downstream speeds are known, we use the standard formulas: b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 These formulas come from solving the two linear equations b + c = downstream speed and b - c = upstream speed. Once we compute b and c, the answer is simply the value of c in km/h.

Step-by-Step Solution:
Step 1: Write the equations for upstream and downstream speeds. b - c = 16. b + c = 24. Step 2: Add the two equations to solve for b. (b - c) + (b + c) = 16 + 24. 2b = 40. b = 40 / 2 = 20 km/h. Step 3: Substitute b into either equation to find c. Using b + c = 24 gives 20 + c = 24. c = 24 - 20 = 4 km/h.
Verification / Alternative check:
We can check using the upstream equation. If b = 20 and c = 4, then b - c = 20 - 4 = 16 km/h, which matches the given upstream speed. Downstream speed with these values is b + c = 20 + 4 = 24 km/h, which also matches the given downstream speed.
Why Other Options Are Wrong:
If c were 6 km/h, then b would be 21 km/h to keep downstream at 27 km/h, which does not match the given data. If c were 5, 3 or 8 km/h, substituting into b - c and b + c would give upstream and downstream speeds that do not equal 16 km/h and 24 km/h respectively.
Common Pitfalls:
A common error is to average the two speeds incorrectly or forget to divide the difference by 2 when computing the current speed. Another pitfall is mixing up which expression gives the boat speed and which gives the current speed, leading to swapped values.
Final Answer:
The speed of the current is 4 kmph.