A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes.

Solution:

Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$.

$\therefore$ Downstream speed $=(9+x) \mathrm{km} / \mathrm{hr}$

Upstream speed $=(9-x) \mathrm{km} / \mathrm{hr}$

Distance covered downstream $=$ Distance covered upstream $=15 \mathrm{~km}$

Total time taken $=3$ hours 45 minutes $=\left(3+\frac{45}{60}\right)$ minutes $=\frac{225}{60}$ minutes $=\frac{15}{4}$ minutes

$\therefore \frac{15}{(9+x)}+\frac{15}{(9-x)}=\frac{15}{4}$

$\Rightarrow \frac{1}{(9+x)}+\frac{1}{(9-x)}=\frac{1}{4}$

$\Rightarrow \frac{9-x+9+x}{(9+x)(9-x)}=\frac{1}{4}$

$\Rightarrow \frac{18}{9^{2}-x^{2}}=\frac{1}{4}$

$\Rightarrow \frac{18}{81-x^{2}}=\frac{1}{4}$

$\Rightarrow 81-x^{2}=72$

$\Rightarrow 81-x^{2}-72=0$

$\Rightarrow-x^{2}+9=0$

$\Rightarrow x^{2}=9$

$\Rightarrow x=3$ or $x=-3$

The value of $x$ cannot be negative; therefore, the speed of the stream is $3 \mathrm{~km} / \mathrm{hr}$.