In a river, water is flowing with a speed of 6 km/hr. A motor boat takes equal…

In boats-and-streams problems, downstream speed equals the boat's speed in still water plus the stream's speed, and upstream speed equals the still-water speed minus the stream's speed. Since time = distance / speed, when two distances are covered in the same time, their distance-to-speed ratios must be equal - this pins down the unknown still-water speed.

Applying this to the given trip:

1. Let the boat's speed in still water be b km/hr. The stream speed is given as 6 km/hr, so the downstream speed is (b + 6) km/hr and the upstream speed is (b - 6) km/hr.

2. The boat covers 78 km downstream in the same time it covers 26 km upstream, so 78/(b + 6) = 26/(b - 6).

3. Cross-multiplying: 78(b - 6) = 26(b + 6), which expands to 78b - 468 = 26b + 156.

4. Solving: 78b - 26b = 156 + 468, so 52b = 624, giving b = 12 km/hr.

5. So the downstream speed is 12 + 6 = 18 km/hr and the upstream speed is 12 - 6 = 6 km/hr.

6. Time for the new trip = 75/18 hours + 15/6 hours = 25/6 hours + 15/6 hours = 40/6 hours = 20/3 hours = 6 hours 40 minutes.

Checking against the original trip: at 18 km/hr, 78 km takes 78/18 = 13/3 hours, and at 6 km/hr, 26 km also takes 26/6 = 13/3 hours - the two times match, confirming the still-water speed of 12 km/hr is correct, so the new trip takes 6 hours 40 minutes.