A boat can travel 78 km upstream and back in a total of 32 hours. It can travel 15 km upstream and 52 km downstream in a total of 9 hours. How much distance will the boat cover in 12 hours in still water?

Question

Accepted Answer

Let the speed of the boat in still water be 'b' km/h and the speed of the stream be 's' km/h.

Upstream speed = (b - s) km/h
Downstream speed = (b + s) km/h

From the first condition:
Time taken to travel 78 km upstream = 78 / (b - s)
Time taken to travel 78 km downstream = 78 / (b + s)
Total time = 32 hours
So, 78/(b - s) + 78/(b + s) = 32

From the second condition:
Time taken to travel 15 km upstream = 15 / (b - s)
Time taken to travel 52 km downstream = 52 / (b + s)
Total time = 9 hours
So, 15/(b - s) + 52/(b + s) = 9

Let x = 1/(b - s) and y = 1/(b + s)

From the equations:
78x + 78y = 32  =>  x + y = 32/78 = 16/39
15x + 52y = 9

From the first equation:
x = (16/39) - y

Substitute in second equation:
15((16/39) - y) + 52y = 9
(240/39) - 15y + 52y = 9
(240/39) + 37y = 9
37y = 9 - (240/39)
37y = (351/39) - (240/39) = 111/39
y = (111/39) / 37 = 111 / (39*37) = 111 / 1443

x = (16/39) - (111/1443) = (592/1443) - (111/1443) = 481/1443

Now,
b - s = 1/x = 1443/481 ≈ 3 km/h
b + s = 1/y = 1443/111 ≈ 13 km/h

Adding:
2b = 3 + 13 = 16
b = 8 km/h (speed of boat in still water)

Distance covered in 12 hours in still water = speed * time = 8 * 12 = 96 km.

Answer

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