X and Y run a 3 km race along a circular course of length 300 m. Their speeds are in the ratio of 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 passing)?

The correct option is 3.

Let's break down the solution step-by-step to understand why the first runner passes the second runner exactly 3 times during the race.

Step 1: Understand the race parameters
The total length of the race is 3 km, which is equal to 3000 m.
The race is run along a circular course of length 300 m.
The speeds of the two runners, X and Y, are in the ratio of 3 : 2. Since X has a higher speed, X is the first runner who will overtake or pass Y.
Let the speed of X be 3v and the speed of Y be 2v, where v is a constant factor of speed.

Step 2: Determine when the first passing (meeting) occurs
X and Y start together from the same point in the same direction.
For X to pass Y for the first time after the start, X must gain a lead of exactly one full round (circle length) over Y.
The circumference of the circular track is 300 m. Therefore, X passes Y when X has run 300 m more than Y.
Let us find the relative speed of X with respect to Y. Since they are running in the same direction, their relative speed is:
$$\text{Relative Speed}=3v-2v=v$$
The time taken for X to gain a lead of 300 m (the first passing) is:
$$t=\frac{300}{v}$$

Step 3: Calculate the distance covered by X at the first passing
During this time $t$, the distance covered by X is:
$$D_X=\text{Speed of X}\times t=3v\times \frac{300}{v}=900\text{m}$$
This means X passes Y for the first time when X has run exactly 900 m.

Step 4: Find the total number of passings during the entire race
Since the race is a continuous run along the same track, X will pass Y every time X covers an additional 900 m.
X passes Y at distances of:
1st passing: 900 m
2nd passing: 900 + 900 = 1800 m
3rd passing: 1800 + 900 = 2700 m
4th passing (if it occurred): 2700 + 900 = 3600 m
Since the total race distance is 3000 m, X finishes the race at 3000 m. The 4th passing would require X to reach 3600 m, which is beyond the finish line of the race.
Therefore, X passes Y at 900 m, 1800 m, and 2700 m. X passes Y exactly 3 times during the 3 km race.