## Bike Rally

You enter a distance bike race that travels between two nearby towns. The first half of the race begins at the first town and takes you to the second town at an average speed of $12$ miles per hour. You return to the first town traveling the same path at an average speed of $8$ miles per hour over the same distance because the return trip was primarily at a slight incline. If the race was completed without a stop, what was your average speed?
Source: mathcontest.olemiss.edu 4/22/2013

SOLUTION
Let $d$ denote the distance in miles from the first town to the second town, $t_1$ the time in hours traveled from the first town to the second town, and $t_2$ the time traveled on the return trip.
$d=12t_1\quad (1)$
$d=8t_2\quad\;(2)$
By definitionĀ the average speed of the round trip is expressed by
$\displaystyle\frac{2d}{t_1+t_2}$
We have
$12t_1=8t_2$
$\displaystyle\frac{t_1}{8}=\frac{t_2}{12}$
$\displaystyle\frac{t_1}{8}=\frac{t_2}{12}=\frac{t_1+t_2}{20}$
$t_1+t_2=\displaystyle\frac{20t_1}{8}$
Average speed of the round trip
$\displaystyle\frac{2d}{t_1+t_2}=\frac{2(12t_1)}{\frac{20t_1}{8}}$
$=\displaystyle\frac{24}{1}\times \frac{8}{20}$
$=\displaystyle\frac{48}{5}$
$=9.6$ miles per hour

Answer: $9.6$ miles per hour