# Jacqueline rows a boat downstream for 96 miles. The return trip upstream took 16 hours longer. ...

## Question:

Jacqueline rows a boat downstream for 96 miles. The return trip upstream took 16 hours longer.

If the current flows at 4 mph, how fast does Jacqueline row in still water?

## Rate of boat

The rate of a boat is a common algebraic word problem. In this problem, the velocity of the boat in upstream, downstream, or in still water is given, then one would calculate other parameters as requested by the problem.

The problem asks for the velocity of the boat in still water. In order to obtain this result, we should take into account the velocity of the boat upstream and downstream.

Recall the formula for rate is:

$$rate\,=\,\dfrac{distance\,traveled}{total\,time}$$

Let

v = speed of the boat in still water

u = speed of the current

t = time

We first compute for the distance traveled by boat downstream and upstream. In downstream, the speed of the boat is much faster because the current is also downstream while in upstream the speed of the boat is much slower because the current is downstream that it added 16 hrs on its return trip.

in downstream:

$$d = (v+u)t$$

in upstream:

$$d = (v-u)(t+16)$$

Substituting d = 96 miles and u = 4mph and simplifying the equations above

$$vt + 4t = 96$$

$$vt + 16v - 4t - 64 = 96$$

Subtracting the upstream equation to the downstream equation yields:

$$(vt + 16v - 4t - 64) - (vt + 4t) = 0$$

$$16v - 8t - 64 = 0$$

$$8t = 16v - 64$$

$$t = 2v - 8$$

Substituting this to the downstream equation:

$$(v+4)(2v-8) = 96$$

$$2v^2 - 32 = 96$$

$$v^2 - 16 = 48$$

$$v^2 = 64$$

$$v = 8\,mph$$

The speed of Jacqueline in still water is {eq}8 {/eq}mph. 