A motorboat whose speed is 18 kmph in still water takes 1 hr 30 min more to go 36 km upstream than to return downstream to the same spot.
Question:

A motorboat whose speed is 18 kmph in still water takes 1 hr 30 min more to go 36 km upstream than to return downstream to the same spot. Find the speed of the stream.

Solution:

Let speed of stream be x km/h.

Given:

Speed of boat = 18 km/h
Distance covered upstream = 36 km
Distance covered downstream = 36 km

It takes $\frac{3}{2}$ hours more to go $36 \mathrm{~km}$ upstream than to return downstream to the same spot

Now, Speed of boat upstream = 18 − x km/h
Speed of boat downstream = 18 + x km/h

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

According to the question,

Time to go upstream $=\frac{3}{2}$ hours $+$ Time to go downstream

Time to go upstream $-$ Time to go downstream $=\frac{3}{2}$

$\frac{36}{18-x}-\frac{36}{18+x}=\frac{3}{2}$

$\Rightarrow \frac{36(18+x)-36(18-x)}{(18+x)(18-x)}=\frac{3}{2}$

$\Rightarrow \frac{648+36 x-648+36 x}{324-x^{2}}=\frac{3}{2}$

$\Rightarrow \frac{72 x}{324-x^{2}}=\frac{3}{2}$

$\Rightarrow \frac{24 x}{324-x^{2}}=\frac{1}{2}$

$\Rightarrow 24 x(2)=324-x^{2}$

$\Rightarrow x^{2}+48 x-324=0$

$\Rightarrow x^{2}+54 x-6 x-324=0$

$\Rightarrow x(x+54)-6(x+54)=0$

$\Rightarrow(x+54)(x-6)=0$

$\Rightarrow x=-54,6$

But $x$ is the speed of stream which is always positive.

Thus, $x=6 \mathrm{~km} / \mathrm{h}$

Hence, the speed of the stream is 6 km/h.

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