Question

The boat goes $30km$ upstream and $44km$ downstream in $10$ hours. In $13$ hours, it can go $40km$ upstream and $55km$ downstream. Determine the speed of stream and that of the boating till water.

• A. Speed of boat $=8km/h$ & Speed of stream $=3km/hr$
• B. Speed of boat $=8km/h$ & Speed of stream $=4km/hr$
• C. Speed of boat $=7km/h$ & Speed of stream $=2km/hr$
• D. Data insufficient

Answer 1

Answer: (A)

The correct option is A (Speed of boat $=8km/h$ & Speed of stream $=3km/hr$)

Let the speed of boat in still water$=xkm/hr$ and
The speed of stream$=ykm/hr$

Speed of boat at downstream,

$\Rightarrow (x+y)km/hr$

Speed of boat at upstream,

$\Rightarrow (x-y)km/hr$

$\because time=\frac{\text{distance}}{\text{speed}}$

Time taken to cover $30km$ upstream $\Rightarrow \frac{30}{x-y}$

Time taken to cover $44km$ downstream$\Rightarrow \frac{44}{x+y}$

According to the first condition,

$\Rightarrow \frac{30}{x-y}=\frac{44}{x+y}=10$

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

Time taken to cover $55km$ downstream $\Rightarrow \frac{55}{x+y}$

According to the second condition,

$\Rightarrow \frac{40}{x-y}=\frac{55}{x+y}=13$

Let $\frac{1}{x-y}=uand\frac{1}{x+y}=v$

$\Rightarrow 30u+44v=\mathrm{10.....}eq1$

$\Rightarrow 40u+55v=\mathrm{13.....}eq2$

Multiplying eq1 by $5$ and eq2 by $4$ and subtract both

$\Rightarrow (150u+220v=50)-(160u+220v=52)$

$\Rightarrow -10u=-2\Rightarrow u=\frac{1}{5}$

put $u=\frac{1}{5}$ in eq1

$\Rightarrow 30\times \frac{1}{5}+44v=10\Rightarrow 44v=4\Rightarrow v=\frac{1}{11}$

$\Rightarrow u=\frac{1}{x-y}=\frac{1}{5}\Rightarrow x-y=\mathrm{5...}eq3$

$\Rightarrow v=\frac{1}{x+y}=\frac{1}{11}\Rightarrow x+y=\mathrm{11...}eq4$

Adding eq3 and eq4, we get

$\Rightarrow x-y+x+y=5+11$
$\Rightarrow 2x=16$

$\therefore x=8$

Put $x=8$ in eq3
$\Rightarrow 8-y=5$
$\Rightarrow y=8-5$

$\therefore y=3$

Hence, the speed of the boat in still water$=8km/hr$

The speed of stream$=3km/hr$