# A speedboat drives 3 times the speed of the current. Thus, it travels downriver for 700 miles 3 hours more than it takes to travel upriver for 325 miles. How do you find the speed of the boat in still water?

Sep 10, 2015

Speed of boat (in still water)$= 12.5 \left(\text{ miles")/(" hour}\right)$

#### Explanation:

Define:
$\textcolor{w h i t e}{\text{XXX}} b$: speed of boat in still water
$\textcolor{w h i t e}{\text{XXX}} c$: speed of current
$\textcolor{w h i t e}{\text{XXX}} {s}_{\mathrm{do} w n}$: speed of boat going downstream
$\textcolor{w h i t e}{\text{XXX}} {s}_{u p}$: speed of boat going upstream
$\textcolor{w h i t e}{\text{XXX}} {t}_{\mathrm{do} w n}$: time to go 700 miles downstream
$\textcolor{w h i t e}{\text{XXX}} {t}_{u p}$: time to go 325 miles upstream

$b = 3 c$ (given)

${s}_{\mathrm{do} w n} = 4 c$ (speed of boat plus speed of current)
${s}_{u p} = 2 c$ (speed of boat minus speed of water)

${t}_{\mathrm{do} w n} = \frac{700}{{s}_{\mathrm{do} w n}} = \frac{700}{4 c}$
${t}_{u p} = \frac{325}{{s}_{u p}} = \frac{325}{2 c}$

${t}_{\mathrm{do} w n} = {t}_{u p} + 3$ (given)

$\frac{700}{4 c} = \frac{325}{2 c} + 3$

$\rightarrow \frac{700}{4} = \frac{325}{2} + 3 c$

$\rightarrow 3 c = \frac{700}{4} - \frac{325}{2} = \frac{700}{4} - \frac{650}{4} = \frac{50}{4} = 12.5$

(and remember that the speed of the boat in still water is $3 c$)