Let the speed of the boat in still water be $x$ km/h. The current's speed is $6$ km/h. The distance for one direction is the same in both cases.

Step 1: Define the downstream speed and time

• Downstream speed = $x + 6$ km/h
• Time taken downstream = 4 hours
• Distance = $4(x + 6)$ km

Step 2: Define the upstream speed and time

• Upstream speed = $x - 6$ km/h
• Time taken upstream = 5 hours
• Distance = $5(x - 6)$ km

Step 3: Equating the distances

Since the distances for the downstream and upstream trips are equal:

$4(x + 6) = 5(x - 6)$

Step 4: Solve for $x$

Expand and simplify:

$4x + 24 = 5x - 30$

$24 + 30 = 5x - 4x$

$x = 54 \, \text{km/h (boat's speed in still water)}.$

Step 5: Calculate the distance

Using the downstream distance formula:

$\text{Distance} = 4(x + 6) = 4(54 + 6) = 4 \times 60 = 240 \, \text{km}.$

Thus, the boat traveled 240 km in one direction.

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Do you want further clarification or have any questions?

Follow-up Questions

1. How would the solution change if the current speed were different?
2. What happens if the times for downstream and upstream trips were reversed?
3. How do we handle cases where the current speed exceeds the boat's still-water speed?
4. Can you derive a general formula for such problems?
5. How would the problem change if both trips took the same time?

Tip: Always ensure consistency between units (e.g., time in hours and speed in km/h) to avoid calculation errors.