Question

A motorboat takes 5 hours to travel 100mi going upstream. The return trip takes 2 hours going downstream. What is the rate of the boat in still water and what is the rate of the current?

Answer 1

The boat went 5hours upstream, let's say it has a "still water" speed rate of "b", it went 100miles... however, going upstream is going against the current, let's say the current has a speed rate of "c"

so, when the boat was going up, it wasn't really going "b" fast, it was going " b - c " fast, because the current was eroding speed from it

now, when coming down, the return trip, well the length is the same, so the distance is also 100miles, it only took 2hrs though, because, the boat wasn't coming down  "b" fast, it was coming down " b + c " fast, because the current was adding speed to it, so it came down quicker

now, recall your d = rt, distance = rate * time

$\bf \begin{array}{lccclll} &distance&rate&time\\ &-----&-----&-----\\ upstream&100&b-c&5\\ downstream&100&b+c&2 \end{array} \\\\\\ \begin{cases} 100=(b-c)5\\ \qquad \frac{100}{5}=b-c\\ \qquad 20=b-c\\ \qquad 20+c=\boxed{b}\\ 100=(b+c)2\\ \qquad 50=b+c\\ \qquad 50=\boxed{20+c}+c \end{cases}$

solve for "c", to see what's the current's speed

what's "b"?  well 20+c = b