Question

Solve the following problems using two variables.
(x) A boat takes 6 hours to travel 8 km upstream and 32 km downstream, and it takes 7 hours to travel 20 km upstream and 16 km downstream. Find the speed of the boat in still water and the speed of the stream.

Answer 1

Let the speed of the boat in still water be x km/h.
Let the speed of the stream be y km/h.
Now, speed of boat in downstream = (x + y) km/h
Also, speed of boat in upstream = (x $-$ y) km/h
According to the question, we have:
$$\begin{gathered} \frac{8}{x-y}+\frac{32}{x+y}=6...\left(1\right) \\ \mathrm{and} \\ \frac{20}{x-y}+\frac{16}{x+y}=7...\left(2\right) \\ \mathrm{Substituting}\frac{1}{x-y}=m\mathrm{and}\frac{1}{x+y}=n,\mathrm{equations}\left(1\right)\mathrm{and}\left(2\right)\mathrm{become}: \\ 8m+32n=6...\left(3\right) \\ 20m+16n=7...\left(4\right) \\ \mathrm{Multiplying}\mathrm{equation}\left(4\right)\mathrm{by}2,\mathrm{we}\mathrm{get}: \\ 40m+32n=14...\left(5\right) \\ \mathrm{Subtracting}\left(3\right)\mathrm{from}\left(5\right),\mathrm{we}\mathrm{get}: \\ 32m=8 \\ \Rightarrow m=\frac{1}{4} \\ \mathrm{Substituting}\mathrm{the}\mathrm{value}\mathrm{of}m\mathrm{in}\left(3\right),\mathrm{we}\mathrm{get}: \\ n=\frac{1}{8} \\ \mathrm{Replacing}\mathrm{the}\mathrm{values}\mathrm{of}m\mathrm{and}\mathit{}n,\mathrm{we}\mathrm{get}: \\ \frac{1}{x-y}=\frac{1}{4} \\ \Rightarrow x-y=4...\left(6\right) \\ \mathrm{and} \\ \frac{1}{x+y}=\frac{1}{8} \\ \Rightarrow x+y=8...\left(7\right) \\ \mathrm{By}\mathrm{adding}\mathrm{equation}\left(6\right)\mathrm{and}\left(7\right),\mathrm{we}\mathrm{get}: \\ x=6 \\ \mathrm{Substituting}x=6\mathrm{in}\mathrm{equation}\left(7\right),\mathrm{we}\mathrm{get}: \\ y=2 \\ \therefore \mathrm{Speed}\mathrm{of}\mathrm{the}\mathrm{boat}\mathrm{in}\mathrm{still}\mathrm{water}=6\mathrm{km}/\mathrm{h} \\ \mathrm{Also},\mathrm{speed}\mathrm{of}\mathrm{the}\mathrm{stream}=2\mathrm{km}/\mathrm{h} \end{gathered}$$