Question

A motor boat took 5 h to travel a distance of

60 km up a river, against the current. The return trip took 3 h. Find the average speed of the boat in still water and the speed of the current.

Answer 1

b = speed of the boat in still water

c = speed of the current

when going Upstream, the boat is not really going "b" fast, is really going slower, is going "b - c", because the current is subtracting speed from it, likewise, when going Downstream the boat is not going "b" fast, is really going faster, is going "b + c", because the current is adding its speed to it.

`{\Large \begin{array}{llll} \underset{distance}{d}=\underset{rate}{r} \stackrel{time}{t} \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{lcccl} &\stackrel{km}{distance}&\stackrel{kmh}{rate}&\stackrel{hrs}{time}\\ \cline{2-4}&\\ Upstream&60&b-c&5\\ Downstream&60&b+c&3 \end{array}\hspace{5em} \begin{cases} 60=(b-c)(5)\\\\ 60=(b+c)(3) \end{cases} \\\\[-0.35em] ~\dotfill`

`\stackrel{\textit{using the 1st equation}}{60=(b-c)5}\implies \cfrac{60}{5}=bc\implies 12=b-c\implies 12+c=b \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{using the 2nd equation}}{60=(b+c)3}\implies \cfrac{60}{3}=b+c\implies 20=b+c\implies \stackrel{\textit{substituting from above}}{20=(12+c)+c} \\\\\\ 20=12+2c\implies 8=2c\implies \cfrac{8}{2}=c\implies \boxed{4=c}~\hfill \stackrel{ 12~~ + ~~4 }{\boxed{b=16}}`