To solve this problem, let's break it down into smaller parts.

First, let's determine the total distance covered by the particle. Since the particle covers half the distance at a speed of 3 m/s and the other half at different speeds, we need to find the total distance.

Let's assume the total distance is d. The first half of the distance would be d/2, and the second half of the distance would also be d/2.

Now, let's calculate the time taken to cover the first half of the distance. We can use the formula: time = distance/speed.

For the first half of the distance, the time taken would be (d/2) / 3 = d/6.

Next, let's calculate the time taken to cover the second half of the distance. We know that the time is divided into two equal intervals, so each interval would take an equal amount of time.

Let's assume the time taken for each interval is t. So, the time taken to cover the second half of the distance would be t + t = 2t.

Using the formula time = distance/speed, we can calculate the time taken for each interval.

For the first interval, the time taken would be (d/2) / 4.5 = d/9.

For the second interval, the time taken would be (d/2) / 7.5 = d/15.

Since the total time taken for the second half is the sum of the times taken for each interval, we have:

2t = (d/9) + (d/15)

Now, let's simplify this equation:

2t = (5d/45) + (3d/45)

2t = (8d/45)

t = (4d/45)

Since t represents the time taken for each interval, the total time taken for the second half would be 2t, which is (8d/45).

Now, let's calculate the total time taken for the entire distance covered by the particle.

The total time would be the sum of the time taken for the first half and the time taken for the second half:

Total time = (d/6) + (8d/45)

Now, let's simplify this equation:

Total time = (15d/90) + (8d/45)

Total time = (23d/90)

Finally, let's calculate the average speed of the particle:

Average speed = total distance / total time

Average speed = d / [(d/6) + (23d/90)]

Average speed = 6d / (d + 23d/15)

Average speed = 6d / (d/15 + 23d/15)

Average speed = 6d / (24d/15)

Average speed = (6d * 15) / (24d)

Average speed = 90d / 24d

Average speed = 3.75 m/s

Therefore, the correct answer is option 'A' - 4.0 m/s.