Circular track race problem Suppose A and B are running a 3km race in a circular track of length 300m. Speeds of A and B are in the ratio 4:3
How often and where would the winner pass the other?
In this situation it is easy to find the time after which the A crosses B for the first time. ( circumference/ relative speed)
Similarly the crossing point can be calculated.
How do I calculate the number of times A crosses B during the whole race?

You have the time between passes.  You can compute the total time $A$ is running by dividing the distance by his speed.  Now divide the running time by the time between passes and throw away the remainder.
