Two friends run a 3 kilometer race along a circular course of length 300 meters.
If their speeds are in ratio 3:2, the number of times the winner passes the other is _____________.

Lets assume two people to be "A" & "B" . It is told that they each had run 3Kms on a 300 m circular track , this implies each had run$$\dfrac{\ 3000\ m}{300\ m}\ =\ 10\ rounds$$ .

Speeds of A:B given as 3:2 . Now, this statement implies that if A takes 100 sec to finish 1 round i.e 300 m circular track, then B should be taking 150 sec to complete 1 round . { Time ratio and Speed ratio are inverse related as distance of each round is constant for both }.

The above deduction can say that if A finishes 3 rounds in 300 sec of time, then B will finish only 2 rounds in the same 300 sec of time. This implies for every 3 rounds which A makes , he will meet/cross B once in his journey.

Therefore, it is given 10 rounds of A, then he will be meeting B : $$\left[\dfrac{\ 10}{3}\right]$$ = 3 times.