Introduction / Context:
This question tests the standard boats-and-streams concept: the speed you observe downstream and upstream is affected by the stream (current). Downstream speed is increased by the current, while upstream speed is decreased by the current. Using these two observed speeds, we can recover the boat's actual speed in still water, which is the base speed of the boat when there is no current at all.
Given Data / Assumptions:
- Downstream (along the current) speed = 14 km/h
- Upstream (against the current) speed = 7 km/h
- Let boat speed in still water = b km/h
- Let speed of current = c km/h
- We need to find b
Concept / Approach:
In boats and streams problems:
- Downstream speed = b + c
- Upstream speed = b - c
If we add the downstream and upstream speeds, the current cancels out:
(b + c) + (b - c) = 2b
So the boat's still-water speed is simply the average of the downstream and upstream speeds:
b = (downstream + upstream) / 2
This is the quickest and most reliable method here.
Step-by-Step Solution:
Step 1: Write the two speed equations.
b + c = 14
b - c = 7
Step 2: Add both equations to eliminate c.
(b + c) + (b - c) = 14 + 7
2b = 21
Step 3: Solve for b.
b = 21 / 2 = 10.5
Verification / Alternative check:
If b = 10.5, then the current speed is c = 14 - 10.5 = 3.5.
Check upstream: b - c = 10.5 - 3.5 = 7, which matches the given upstream speed.
So the computed still-water speed is consistent with both directions.
Why Other Options Are Wrong:
9.5 would imply current c = 14 - 9.5 = 4.5, giving upstream 9.5 - 4.5 = 5, not 7.
8.5 would imply current c = 5.5, giving upstream 3, not 7.
11 would imply current c = 3, giving upstream 8, not 7.
7.5 would imply current c = 6.5, giving upstream 1, not 7.
Common Pitfalls:
Many learners incorrectly subtract the two speeds and divide by 2, which actually gives the current speed, not the boat speed.
Another common mistake is to assume the still-water speed equals the downstream speed, ignoring the effect of the current.
Final Answer:
The speed of the boat in still water is 10.5 km/h.
Discussion & Comments