Question

A motor boat can travel $30$ km upstream and $28$ km downstream in $7$ hours , it can travel $21$ km upstream and return in $5hours$ find the speed of the boat in still water and the speed of the stream.

Answer 1

Let speed of boat in still water $=xkm/hr$

and the speed of the stream $=ykm/hr$

Speed of motor boat upstream $=(x-y)km/hr$

Speed of motor boat downstream $=(x+y)km/hr$

Case I : Time taken by motor boat in $30km$ upstream $=\frac{30}{x-y}hr$

Time taken by motor boat in $28km$ downstream $=\frac{28}{x+y}hr$

$\therefore \frac{30}{(x-y)}+\frac{28}{(x+y)}=7$

$\Rightarrow 2[\frac{15}{(x-y)}+\frac{14}{(x+y)}]=7$

$\Rightarrow \frac{15}{x-y}+\frac{14}{x+y}=\frac{7}{2}$ ___(i)

Case II : Time taken by motor boat in $21km$ upstream $=\frac{21}{x-y}hr$

Time taken by motor boat to return $21km$ downstream $=\frac{21}{x+y}hr$

$\therefore \frac{21}{x-y}+\frac{21}{x+y}=5$

$\Rightarrow 21[\frac{1}{x-y}+\frac{1}{x+y}]=5$

$\Rightarrow \frac{1}{x-y}+\frac{1}{x+y}=\frac{5}{21}$ ____(ii)

$\frac{15}{x-y}+\frac{14}{x+y}=\frac{7}{2}$ [From (i)]

As equations (both) are symmetric to $(x-y)$ and $(x+y)$ so we can eliminate either $(x-y)$ or $(x+y)$

Multiplying (ii) by $14$, we get

$\frac{14}{(x-y)}+\frac{14}{(x+y)}=\frac{70}{21}$ ___(iii)

$\frac{15}{(x-y)}+\frac{14}{x+y}=\frac{7}{2}$ [From (i)]

$-$ $-$ $-$

$\overline{\frac{14}{(x-y)}-\frac{15}{(x-y)}=\frac{10}{3}-\frac{7}{2}}$ [Subtracting (i) from (iii)]

$\Rightarrow \frac{14-15}{(x-y)}=\frac{20-7\times 3}{3\times 2}$

$\Rightarrow \frac{-1}{(x-y)}=\frac{-1}{6}$ ___(iv)

$\Rightarrow (x-y)=6$

Now, substituting $x-y=6$ in (ii), we have

$\frac{1}{(x-y)}+\frac{1}{(x+y)}=\frac{5}{21}$

$\Rightarrow \frac{1}{6}+\frac{1}{(x+y)}=\frac{5}{21}$

$\Rightarrow \frac{1}{(x+y)}=\frac{5}{21}-\frac{1}{6}$

$\Rightarrow \frac{1}{(x+y)}=\frac{2\times 5-7\times 1}{3\times 7\times 2}$

$\Rightarrow \frac{1}{(x+y)}=\frac{3}{42}$

$\Rightarrow \frac{1}{(x+y)}=\frac{1}{14}$

$\Rightarrow x+y=14$ __(v)

$x-y=6$ [From (iv)]

$\overline{2x=20}$ [Subtracting (iv) from (v)]

$\Rightarrow x=10km/hr$

Now, $x+y=14$ [from (v)]

$\Rightarrow 10+y=14$

$\Rightarrow y=4km/hr$

Hence, the speed of motorboat and stream are $10km/hr$ and $4km/hr$ respectively.