A motorboat can go at 8 km/h in still water. It takes three times as long to go a fixed distance against the current as it takes to go the same distance with the current. What is the speed (in km/h) of the current?

Introduction / Context:
This problem describes a motorboat whose speed in still water is known. The boat takes three times as long to go a certain distance against the current as it takes to go the same distance with the current. We are asked to find the speed of the current. This is a typical algebraic boats and streams question that uses the relationship between time, distance and effective speed in upstream and downstream directions.

Given Data / Assumptions:

• Speed of the boat in still water b = 8 km/h.
• Let the speed of the current be c km/h.
• Downstream speed = b + c = 8 + c km/h.
• Upstream speed = b - c = 8 - c km/h.
• The time taken upstream is three times the time taken downstream for the same distance.
• We must find c.

Concept / Approach:
If a fixed distance d is covered in time t1 upstream and t2 downstream, then: t1 = d / (b - c) t2 = d / (b + c) We are told that the upstream time is three times the downstream time, so t1 = 3t2. After substituting the formulas for t1 and t2 and cancelling the common distance d, we get an equation involving only b and c. Since b is known, we can solve this equation to obtain c, the speed of the current.

Step-by-Step Solution:
Step 1: Let the one way distance be d km. Upstream time = d / (8 - c). Downstream time = d / (8 + c). Step 2: Use the condition that upstream time is three times downstream time. d / (8 - c) = 3 * d / (8 + c). Step 3: Cancel d from both sides. 1 / (8 - c) = 3 / (8 + c). Step 4: Cross multiply to remove fractions. 8 + c = 3(8 - c). 8 + c = 24 - 3c. Step 5: Collect like terms. c + 3c = 24 - 8. 4c = 16. c = 16 / 4 = 4 km/h.
Verification / Alternative check:
With c = 4 km/h, downstream speed = 8 + 4 = 12 km/h and upstream speed = 8 - 4 = 4 km/h. For a distance of 12 km, downstream time = 12 / 12 = 1 hour. Upstream time = 12 / 4 = 3 hours, which is exactly three times the downstream time. This confirms that c = 4 km/h is consistent with the conditions.
Why Other Options Are Wrong:
If c were 6 km/h, upstream speed would be 2 km/h and downstream speed 14 km/h, which would not give an upstream time exactly three times the downstream time for the same distance. Similarly, c values of 3, 2 or 1 km/h lead to different time ratios that do not equal 3.
Common Pitfalls:
One common mistake is to reverse the time ratio and write downstream time as three times the upstream time, which leads to the wrong equation. Another error is not cancelling the distance d properly and trying to keep it as an extra variable, which complicates the algebra unnecessarily.
Final Answer:
The speed of the current is 4 kmph.