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?
 A: 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.
A: Describe the orbits of the two runners by
$$b(t)= e^{2\pi i t}\quad(0\leq t\leq 10), \qquad a(t)=e^{2\pi i(4t/3)}\quad(0\leq t\leq 7.5)\ .$$
We have a coincidence on the track when $t-{4\over3}t\in{\mathbb Z}$, i.e., when $t\in3{\mathbb Z}$. This happens at the common start $t=0$, then at $t=3$ and $t=6$. It follows that there are two actual passings, both occurring at the starting line.
A: Suppose $A$ runs at 4m/unit of time (Call it a "u") and $B$ runs at 3m/u.
In $t$ units of  time runner $A$ will have run $4t$ meters and runner be will have run $3t$ meters.  If they are at the same point in the track then $4t = 3t + n*300$ were $n$ is how many more times around the track $A$ has run.  $n$ is also the number of times they have meet.
$4t = 3t + n*300$
$t = n*300$ 
The racers will be at the same point every 300 units of time. 
How long does the race last?  Well, racer $A$ will win so $4T = 3000$ meters so $T = 750u$.  
How many times will Player A outpace B if the meet every 300 u?  $750u/300u = 2.5$  So 2 times. (must be  whole number).
Another way to look at it is:  Every 4 laps A runs, B runs 3.  So in 4 laps A has met with B once.  And that time happens to be one an even lap! A runs 10 laps so he will meet with B, 2 times. (and be half a lap ahead at the end).
Another note:  In 300 units, $A$ will have run 1200 meters or 3 laps exactly.  $B$ will have run 900 meters or 3 laps exactly.
