Question

A boat takes $6$ hours to travel $16$ km upstream and $24$ km downstream; but it takes $13$ hours to travel $36$ km upstream and $48$ km downstream. Find the speed of the boat in still water and the speed of the stream.

Answer 1

Let speed of boat in still water be $ukm/hr$

Let Speed of stream be $vkm/hr$

Upstream Speed$=(u-v)km/hr$

Downstream Speed$(u+v)km/hr$

$\frac{16}{u-v}+\frac{24}{u+v}=6$

$\Rightarrow \frac{16u+16v+24u-24v}{u^2-v^2}=6$

$\Rightarrow \frac{40u-8v}{u^2-v^2}=6⟶\left(i\right)$

$\Rightarrow \frac{36}{u-v}+\frac{48}{u+v}=13$

$=\frac{36u+36v+48u-48v}{u^2-v^2}=13$

$\Rightarrow \frac{84u-12v}{u^2-v^2}=13⟶\left(ii\right)$

$\left(i\right)÷\left(ii\right)$

$=\frac{40u-8v}{84u-12v}=\frac{6}{13}$

$520u-104v=504u-72v$

$16u=32v$

$u=2v$

Putting $u=2v$ in equation$\left(i\right)$

$\frac{40\left(2v\right)-8v}{{\left(2v\right)}^2-v^2}=6$

$\frac{80v-8v}{3v^2}=6$

$72=18v$

$v=4km/hr$

$u=8km/hr$