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

Dear student,

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;
$$\begin{gathered} 32u+36v = 7 ...\left(\mathrm{iii}\right) \\ 40\mathrm{u}+48\mathrm{v} = 9 ...\left(\mathrm{iv}\right) \end{gathered}$$
Multiplying (iii) by 5 and (iv) by 4 and then subtracting (iv) from (iii) we get;
$$\begin{gathered} 160u+180v-160u-192v = 35-36 \\ \Rightarrow -12v = -1 \\ \Rightarrow v = \frac{1}{12} \\ \mathrm{substituting} \mathrm{back} v = \frac{1}{y}; \mathrm{we} \mathrm{get}; y = 12 \end{gathered}$$
Putting y = 12 in (i) we get;
$$\begin{gathered} \frac{32}{x}+\frac{36}{12} = 7 \\ \Rightarrow \frac{32}{x} = 4 \\ \Rightarrow x = \frac{32}{4} = 8 \end{gathered}$$
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.