Abicyclist 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?

We know that the first cyclist has traveled 24.2 km at a speed of 10 km/h. Lets use the equation  to find the time of the first cyclist:


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  represents the time of our first cyclist, the time of our second cyclist will be .

Now we are going to use the equation  to relate the speeds and times of the two cyclist:
For the first one:
equation (1)
For the second one:
equation (2)

Replace (1) in (2)

The only thing left now is solve for :





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