X and Y run a 3 km race along a circular course of length 300 m. Their speeds are in the ratio 3:2 . If they start together in the same direction, how many times would the first one pass the other (the start-off is not counted as passing)?

To determine how many times the first runner (X) passes the other (Y), we need to consider their relative speed and the total distance covered.

1. Relative Speed: X's speed is 3 units, Y's speed is 2 units. Since they run in the same direction, X gains on Y at a relative speed of (3 - 2) = 1 unit.
2. Distance for one pass: X passes Y when X gains one full lap on Y. The length of one lap is 300 m.
3. Total race distance: The race is 3 km = 3000 m.
4. Time for X to finish the race: X runs at 3 units of speed. To cover 3000 m, X takes a time proportional to 3000/3 = 1000 units.
5. Time for X to gain one lap: X gains one lap (300 m) at a relative speed of 1 unit. This takes a time proportional to 300/1 = 300 units.
6. Number of passes: Divide the total "time" X is running by the "time" it takes for X to gain one lap. Number of passes = (Time for X to finish) / (Time for X to gain one lap) Number of passes = 1000 / 300 = 10/3 = 3.33...

Since a pass means completing a full lap ahead, we take the integer part. X passes Y 3 times. The start-off is not counted, and our calculation counts distinct instances of X gaining a full lap.

The final answer is B) 3.