Question

A boat goes 80 km upstream and 120 km downstream in 20 hours. In 15 hours, it can go 48 km upstream and 108 km down-stream. Determine the speed of the boat in still water and that of the stream. [4 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

Let speed of boat in still water be $xkm/hr$ and speed of stream be $ykm/hr$

Hence speed of upstream $=x-y$

Speed downstream $=x+y$

Time taken to cover 80 km upstream $=\frac{80}{x-y}$

Time taken to cover 120 km downstream $=\frac{120}{x+y}$

Total time taken is = 20 hrs

$\Rightarrow \frac{80}{x-y}+\frac{120}{x+y}=20$

$\Rightarrow 80u+120v=20$ where $\frac{1}{x-y}=u,\frac{1}{x+y}=v$

$\Rightarrow 4u+6v=1\dots \dots \left(i\right)$

Similarly,

$\Rightarrow \frac{48}{x-y}+\frac{108}{x+y}=15$

$\Rightarrow 48u+108v=15$

$\Rightarrow 16u+36v=5\dots \dots \left(ii\right)$

Solving (i) and (i) we get,

$4u+6v=1\Rightarrow u=\frac{1-6v}{4}$

Substituting value of u in (ii)

$16u+36v=5$

$\Rightarrow 16\left(\frac{1-6v}{4}\right)+36v=5$

$\Rightarrow 4-24v+36v=5$

$\therefore v=\frac{1}{12}$

Now, $u=\frac{1-6v}{4}$

$\Rightarrow u=\frac{1}{8}$

$\therefore v=\frac{1}{12}\Rightarrow \frac{1}{x+y}=\frac{1}{12}$

$\Rightarrow x+y=12\dots \dots \left(iii\right)$

$u=\frac{1}{8}\Rightarrow \frac{1}{x-y}=\frac{1}{8}$

$\Rightarrow x-y=8\dots \dots \left(iv\right)$

Solving (iii) and (iv) we get,

$x=10,y=2$

$\therefore$ Speed of boat in still water is $10km/hr$

And speed of stream $2km/hr$