A motor boat went down the river for $14$ km and then upstream for $9$ km, having covered the whole way in five hours. Find the speed of the river flow if the speed of the boat in still water is $5$ kmph.
Speed of boat in still water $= u = 5$ kmph
Let speed of river flow $= v$ kmph
Then, speed of boat down-stream $= s_d = \left(u + v\right)$ kmph $= \left(5 + v\right)$ kmph
and speed of boat up-stream $= s_{up} = \left(u - v\right)$ kmph $= 5 - v$ kmph
Distance covered by the boat down-stream $= d_d = 14$ km
Time for which the boat went down-stream $= t_d = \dfrac{d_d}{s_d} = \dfrac{14}{5 + v}$ hr
Distance covered by the boat up-stream $= d_{up} = 9$ km
Time for which the boat went up-stream $= t_{up}= \dfrac{d_{up}}{s_{up}} = \dfrac{9}{5 - v}$ hr
Time taken by the boat to go up-stream and down-stream
$= t_{total} = t_{up} + t_{d} = \dfrac{9}{5-v} + \dfrac{14}{5 + v}$
Given: $\;$ $t_{total} = 5$ hr
$\implies$ $\dfrac{9}{5 - v} + \dfrac{14}{5 + v} = 5$
i.e. $\;$ $\dfrac{45 + 9v + 70 - 14v}{25 - v^2} = 5$
i.e. $\;$ $115 - 5v = 125 - 5v^2$
i.e. $\;$ $5v^2 - 5v - 10 = 0$
i.e. $\;$ $v^2 - v - 2 = 0$
i.e. $\;$ $\left(v - 2\right) \left(v + 1\right) = 0$
i.e. $\;$ $v = 2$ $\;$ or $\;$ $v = -1$
$\because \;$ the speed of river flow cannot be negative
$\implies$ $v = -1$ is not a valid solution
$\therefore \;$ Speed of river flow $= v = 2$ kmph