Sophia and Tyler are running around a 400-meter track. They start running from the same place, at the same time. Sophia runs at a speed of 5 m/s and Tyler runs at a speed of 4 m/s. How long will it take Sophia to lap Tyler?

We will call:

S: Runner Sophia.
T: Runner Tyler.

At the beginning the distance between both runners is 400m

At t = 1s Sophia runs 5m and Tyler 4m
At t = 2s Sophia runs 10m and Tyler 8m

and so on.

So, for example at t = 1s, the distance between both runner is:

D = 400m - 5m(the advantage of Sophia) + 4m(what Tyler recovers)

So, we can find a general equation for this matter. Knowing that the distance, velocity and time are related by this equation:

[tex]d=vt[/tex]

Then, the distance between both runners, at any time, is given by:

[tex]D = 400-5t+4t[/tex]

So we need to find the how long it will take Sophia to lap Tyler. This happens when D = 0:

[tex]0 = 400 - 5t+4t[/tex]
∴ [tex]400-t = 0[/tex]
∴ [tex]t = 400s[/tex]