Question

A bicyclist heads east at 10km/h. After she has traveled 24.2 kilometers, another cyclist sets out in the same direction going 30km/h. About how long will it take the second cyclist to catch up to the first cyclist?

Answer 1

We know that the first cyclist has traveled 24.2 km at a speed of 10 km/h. Lets use the equation $time= \frac{distance}{speed}$ to find the time of the first cyclist:
$t= \frac{24.2}{10}$
$t=2.42$
So, we know that she has traveled for 2.42 hours.
 
We also know that after she has traveled for 2.42 hours another cyclist sets out in the same direction, so if $t$ represents the time of our first cyclist, the time of our second cyclist will be $t-2.42$.

Now we are going to use the equation $distance=(speed)(time)$ to relate the speeds and times of the two cyclist:
For the first one:
$distance=10t$ equation (1)
For the second one:
$distance=30(t-2.42)$ equation (2)

Replace (1) in (2)
$10t=30(t-2.42)$
The only thing left now is solve for $t$:
$10t=30t-72.6$
$-20t=-72.6$
$t= \frac{-72.6}{-20}$
$t=3.63$

We can conclude that the second cyclist will catch up the first cyclist after 3.63 hours.