A man rowing at the rate of 5 km an hour in still water takes thrice as much time in going 40 km up the river as in going 40 km down. Find the rate at which the river flows.

Let the rate of flow of the river be x km/h.

The man's speed in still water = 5 km/h.

The effective speeds of the man while rowing upstream and downstream are:

Upstream speed = (5 - x) km/h

Downstream speed = (5 + x) km/h

The time taken to row 40 km upstream:

t_up = $\dfrac{40}{5 - x}$

The time taken to row 40 km downstream:

t_down = $\dfrac{40}{5 + x}$

It is given that the time taken upstream is three times the time taken downstream:

⇒ t_up = 3t_down

$⇒ \dfrac{40}{5 - x} = 3 \times \dfrac{40}{5 + x}\\[1em] ⇒ \dfrac{40}{5 - x} = \dfrac{120}{5 + x}\\[1em] ⇒ 40 \times (5 + x) = 120 \times (5 - x)\\[1em] ⇒ 200 + 40x = 600 - 120x\\[1em] ⇒ 120x + 40x = 600 - 200\\[1em] ⇒ 160x = 400\\[1em] ⇒ x = \dfrac{400}{160}\\[1em] ⇒ x = 2.5$

Hence, the rate at which the river flows = 2.5 km/h.