Question

A boat covers $32km$ upstream and $36km$ downstream in 7 hours. Also, it covers $40km$ upstream and $48km$ 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 $xkm/hr$ and the speed of the stream but $ykm/hr$. Then
Speed upstream $=(x-y)km/hr$
Speed downstream $=(x+y)km/hr$
Now, Time taken to cover $32km$ upstream $=\frac{32}{x-y}hrs$
Time taken to cover $36km$ downstream $=\frac{36}{x+y}hrs$
But, total time of journey is 7 hours.
$$\therefore \frac{32}{x-y}+\frac{36}{x+y}=7..\text{(i)}$$
Time taken to cover $40km$ upstream $=\frac{40}{x-y}$
Time taken to cover $48km$ downstream $=\frac{48}{x+y}$
In this case, total time of journey is given to be 9 hours.
$$\therefore \frac{40}{x-y}+\frac{48}{x+y}=9\left(ii\right)$$
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)
$$40u+48v=9\Rightarrow 40u-48v-9=0..\text{(iv)}$$
Solving these equations by cross-multiplication, we get
$$\begin{array}{c}\frac{u}{36\times -9-48\times -7}=\frac{-v}{32\times -9-40\times -7}=\frac{1}{32\times 48-40\times 36}\\ \Rightarrow \frac{u}{-324+336}=\frac{-v}{-288+280}=\frac{1}{1536-1440}\\ \Rightarrow \frac{u}{12}=\frac{v}{8}=\frac{1}{96}\\ \Rightarrow u=\frac{12}{96}\text{and}v=\frac{8}{96}\end{array}$$
$\Rightarrow u=\frac{1}{8}$ 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 $=10km/hr$ and Speed of the stream $=2km/hr$.