# Two bicyclists ride in opposite directions. The speed of the first bicyclist is 5miles per hour faster than the second. After 2hours they are 70miles apart. How do you find their rates?

Oct 2, 2017

$20$ miles per hour and $15$ miles per hour.
(see below for method of solution)

#### Explanation:

Suppose the second bicyclist is traveling at color(blue)(k miles per hour
(which implies the the first is traveling at $\textcolor{g r e e n}{k + 5}$ miles per hour.

The distance between them is increasing at the rate of
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{k} + \textcolor{g r e e n}{k + 5} = \textcolor{m a \ge n t a}{2 k + 5}$ miles per hour.

If after $\textcolor{red}{2}$ hours they are $\textcolor{b r o w n}{70}$ miles apart:
color(white)("XXX")(color(magenta)(2k+5)" miles")/("hour")xxcolor(red)2" hours"=color(brown)(70)" miles"

Simplifying
color(white)("XXX")4color(blue)k+10" miles"=color(brown)70" miles"

$\textcolor{w h i t e}{\text{XXX}} 4 \textcolor{b l u e}{k} = 60$

$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{k} = 15$

That is, the second bicyclist is traveling at $15$ miles per hour
and the first bicyclist is traveling at $15 + 5 = 20$ miles per hour.