Question

A motor boat whose speed in still water is 9 km/hr, goes 15 km downstream and comes back to the same spot, in a total time of 3 hours 45 minutes. Find the speed of the stream.

Answer 1

Let speed of stream be x km/h.

Given:

Speed of boat = 9 km/h
Distance covered upstream = 15 km
Distance covered downstream = 15 km
Total time taken = 3 hours 45 minutes = $\frac{15}{4}\mathrm{hours}$

Now, Speed of boat upstream = 9 − x km/h
Speed of boat downstream = 9 + x km/h

$\frac{\mathrm{Distance}}{\mathrm{Speed}}=\mathrm{Time}$

According to the question,

$$\begin{gathered} \frac{15}{9+x}+\frac{15}{9-x}=\frac{15}{4} \\ \Rightarrow \frac{15\left(9-x\right)+15\left(9+x\right)}{\left(9+x\right)\left(9-x\right)}=\frac{15}{4} \\ \Rightarrow \frac{135-15x+135+15x}{81-x^2}=\frac{15}{4} \\ \Rightarrow \frac{135+135}{81-x^2}=\frac{15}{4} \\ \Rightarrow \frac{270}{81-x^2}=\frac{15}{4} \\ \Rightarrow \frac{18}{81-x^2}=\frac{1}{4} \\ \Rightarrow 18\left(4\right)=81-x^2 \\ \Rightarrow 72=81-x^2 \\ \Rightarrow x^2=9 \\ \Rightarrow x=\pm 3 \\ \mathrm{But}x\mathrm{is}\mathrm{the}\mathrm{speed}\mathrm{of}\mathrm{stream}\mathrm{which}\mathrm{is}\mathrm{always}\mathrm{positive}. \\ \mathrm{Thus},x=3\mathrm{km}/\mathrm{h} \end{gathered}$$

Hence, the speed of the stream is 3 km/h.