For the first time, they will meet each other, when the sum of the distances they cover will be exactly the length of the track


    . + . = 108  meters,   or


    ( + ). = 108

    9t = 108

     t =  = 12 seconds.


So, for the first time they will meet each other in 12 seconds.


After that, they are in the same situation, as originally: they start from the common point and move 

in opposite directions, with the same rates as before.  Hence, they meet each other again in 12 seconds next time.


Now it should be clear to you that they will meet each other EVERY 12 seconds.


So, to get the answer, you should divide 45 minutes by 12 seconds -- in this way you will get 


    45 minutes divided by 12 seconds = 45 minutes divided by  of the minute = 45*5 = 225 times.


ANSWER.  225 times. 

         We do not count the starting point.

         We count their last meeting at the end of the 45-th minute.