Question

A long distance runner starts at the beginning of a trail and runs at a rate of 4 miles per hour. Three hours later, a cyclist starts at the beginning of the trail and travels at a rate of 19 miles per hour. What is the amount of time that the cyclist travels before overtaking the runner?
Do not do any rounding.

Answer 1

After $h$ hours, the distance the runner went is
$4h$
and the distance the cyclist went is
$19(h-3)$

Set the two equations equal and solve for $h$:
$4h=19(h-3)$
$\Rightarrow 4h=19h-57$
$\Rightarrow -15h=-57$
$\Rightarrow h=\frac{57}{15}$
$\Rightarrow h=\frac{19}{5}$
$\Rightarrow h=3.8$

The cyclist traveled $h-3$ hours, which is 0.8 hours