Question

A boat covers $32$ km upstream and $36$ km downstream in $7$ hours. Also, it covers $40$ km upstream and $48$km downstream in $9$ hours. Find the speed of the boat in still water and that of the stream.

Answer 1

Let the speed of the boat in still water be x km/hr and the speed of the stream but y km/hr. Then,

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

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

Now, Time taken to cover $32$km upstream $=\frac{32}{x-y}$ hrs

Time taken to cover $36$ km downstream $=\frac{36}{x+y}$ hrs

But, total time of journey is $7$ hours.

$\therefore \frac{32}{x-y}+\frac{36}{x+y}=7$ ..(i)

Time taken to cover $40$km upstream $=\frac{40}{x-y}$

Time taken to cover $48$ km downstream $=\frac{48}{x+y}$

In this case, total time of journey is given to be $9$ hours.

$\therefore \frac{40}{x-y}+\frac{48}{x+y}=9$ (ii)

Putting $\frac{1}{x-y}=u$ and $\frac{1}{x+y}=v$ in equations (i) and (ii), we get

$32u+36v=7\Rightarrow 32u-36v-7=0$ ..(iii)

$40u+48v=9\Rightarrow 40u-48v-9=0$ ..(iv)

Solving these equations by cross-multiplication, we get

$\frac{u}{36\times -9-48\times -7}=\frac{-v}{32\times -9-40\times -7}=\frac{1}{32\times 48-40\times 36}$

$\Rightarrow \frac{u}{-324+336}=\frac{-v}{-288+280}=\frac{1}{1536-1440}$

$\Rightarrow \frac{u}{12}=\frac{v}{8}=\frac{1}{96}$

$\Rightarrow u=\frac{12}{96}$ and $v=\frac{8}{96}$

$\Rightarrow u=\frac{1}{8}$ and $v=\frac{1}{12}$

Now, $u=\frac{1}{8}\Rightarrow \frac{1}{x-y}=\frac{1}{8}\Rightarrow x-y=8$ ..(v)

and, $v=\frac{1}{12}\Rightarrow \frac{1}{x+y}=\frac{1}{12}\Rightarrow x+y=12$ ..(vi)

Solving equations (v) and (vi), we get $x=10$ and $y=2$

Hence, Speed of the boat in still water $=10$ km/hr

and Speed of the stream $=2$km/hr.