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A boat covers 32 km upstream and 36 km downstream in 7 hours. Also it covers 40 km upstream and 48 km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.

(CBSE handbook question)

Thanks

Regards

Suppose the speed of upstream be

*x*km/h and that of downstream be

*y*km/h.

When the boat goes 32 km upstream and 36 km downstream in 7 hours.

$\frac{32}{x}+\frac{36}{y}=7...\left(\mathrm{i}\right)$

When the boat goes 40 km upstream and 48 km downstream in 9 hours.

$\frac{40}{x}+\frac{48}{y}=9...\left(\mathrm{ii}\right)$

substituting $\frac{1}{x}=u\mathrm{and}\frac{1}{y}=v$ in (i) and (ii) we have;

$32u+36v=7...\left(\mathrm{iii}\right)\phantom{\rule{0ex}{0ex}}40\mathrm{u}+48\mathrm{v}=9...\left(\mathrm{iv}\right)\phantom{\rule{0ex}{0ex}}$

Multiplying (iii) by 5 and (iv) by 4 and then subtracting (iv) from (iii) we get;

$160u+180v-160u-192v=35-36\phantom{\rule{0ex}{0ex}}\Rightarrow -12v=-1\phantom{\rule{0ex}{0ex}}\Rightarrow v=\frac{1}{12}\phantom{\rule{0ex}{0ex}}\mathrm{substituting}\mathrm{back}v=\frac{1}{y};\mathrm{we}\mathrm{get};y=12$

Putting

*y*= 12 in (i) we get;

$\frac{32}{x}+\frac{36}{12}=7\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{32}{x}=4\phantom{\rule{0ex}{0ex}}\Rightarrow x=\frac{32}{4}=8$

So the speed of upstream is 8 km/h and the speed of downstream is 12 km/h.

So the speed of the boat in still water = $\frac{1}{2}\left(\mathrm{speed}\mathrm{of}\mathrm{upstream}+\mathrm{speed}\mathrm{of}\mathrm{downstream}\right)=\frac{1}{2}\left(8+12\right)=10\mathrm{km}/\mathrm{h}$

Regards.

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