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)?

Correct Option (2)

To determine the number of times the first runner (X) passes the second runner (Y), we analyze their relative movement on the circular track.

• The total race distance is 3 km, which is 3000 m.
• The length of the circular track is 300 m.
• The speeds of X and Y are in the ratio 3:2.

Since X is faster than Y and they run in the same direction, X will pass Y when X gains one full lap on Y. Let the speeds of X and Y be 3v and 2v, respectively.

• When X completes 3 laps, X covers a distance of 3 * 300 m = 900 m.
• In the same amount of time, Y, with a speed ratio of 2/3 compared to X, covers a distance of (2/3) * 900 m = 600 m, which is 2 laps.
• At this point, X has gained 900 m - 600 m = 300 m on Y, which is exactly one full lap.
• Therefore, X passes Y once for every 3 laps X completes.

Now, we calculate the total number of laps completed by X during the race:

• Total laps by X = Total race distance / Track length = 3000 m / 300 m = 10 laps.

The number of times X passes Y is the total laps completed by X divided by the number of laps X completes per pass:

• Number of passes = 10 laps / 3 laps per pass = 3.33...

Since only complete passes are counted, X passes Y 3 times during the race. The start-off is not counted as passing.

Incorrect Options:

Options 1 (2), 3 (4), and 4 (5) are incorrect because they do not reflect the accurate calculation of passes based on the relative speeds and the total distance covered on the circular track. The method of calculating the number of laps gained by the faster runner over the slower runner, divided by the track length, yields an integer result of 3 complete passes, with the fourth pass not being fully completed by the end of the race.