A boat can go 3.6 km upstream and 5.4 km downstream in 54 minutes, while it can go 5.4 km upstream and 3.6 km downstream in 58.5 minutes. The time (in minutes) taken by the boat in going 10 km downstream is:

Let the speed of speed of stream be u and speed of boat in still water be v.

Speed of boat in upstream = u - v

Speed of boat in downstream = u + v

A boat can go 3.6 km upstream and 5.4 km downstream in 54 minutes so,

Time = distance/speed

$$\frac{3.6}{u - v} + \frac{5.4}{u + v} = \frac{54}{60}$$

$$\frac{3.6}{u - v} + \frac{5.4}{u + v} = \frac{9}{10}$$

$$\frac{36}{u - v} + \frac{54}{u + v} = 9$$

4(u + v) + 6(u - v) = $$u^2 - v^2$$

10u - 2v = $$u^2 - v^2$$ ---(1)

Boat can go 5.4 km upstream and 3.6 km downstream in 58.5 minutes so,

$$\frac{5.4}{u - v} + \frac{3.6}{u + v} = \frac{58.5}{60}$$

$$\frac{54}{u - v} + \frac{36}{u + v} = \frac{585}{60}$$

$$\frac{6}{u - v} + \frac{4}{u + v} = \frac{13}{12}$$

72(u + v) + 48(u - v) = 13($$u^2 - v^2$$)

120u + 24v = 13($$u^2 - v^2$$)

Put the value of eq(1),

120u + 24v = 13 $$\times (10u - 2v)$$

120u + 24v = 130u - 26v

10u = 50v

u = 5v ---(2)

put the value of u in eq(1),

50v - 2v = $$25v^2 - v^2$$

48v = 24v^2

v = 2

put the value of v in eq(2),

u = 5$$\times$$ 2 = 10

Speed of boat in downstream = u + v = 10 + 2 = 12

The time taken by the boat in going 10 km downstream = distance/speed = 10/12 hours = 60 $$\times$$ 10/12 = 50 min