Three cyclists start riding from the exact same starting point along a 2.5 km circular velodrome track at speeds of 8 km/h, 10 km/h, and 12 km/h, respectively. If they continuously ride during a 6-hour period from 9:00 AM to 3:00 PM, how many times will all three of them meet simultaneously at the starting point (excluding their initial start at 9:00 AM)?

The correct option is 5 times.

Step 1: Calculate the time taken by each cyclist to complete one full lap.

The length of the circular velodrome track is 2.5 km.

The formula for time taken to complete one round is:

$\text{Time}=\frac{\text{Distance}}{\text{Speed}}$

Calculating the time taken by each cyclist to complete 1 full lap:

• Time taken by Cyclist 1 ($t_1$):

$t_1=\frac{2.5}{8}=\frac{5}{16}\text{hours}$

• Time taken by Cyclist 2 ($t_2$):

$t_2=\frac{2.5}{10}=\frac{1}{4}\text{hours}$

• Time taken by Cyclist 3 ($t_3$):

$t_3=\frac{2.5}{12}=\frac{5}{24}\text{hours}$

Step 2: Find the time interval for simultaneous meeting at the starting point.

All three cyclists will meet together at the starting point after a time interval equal to the Least Common Multiple (LCM) of their individual lap completion times.

To find the LCM of fractions, we use the formula:

$\text{LCM of fractions}=\frac{\text{LCM of Numerators}}{\text{HCF of Denominators}}$

Applying the values:

$\text{LCM}\left(\frac{5}{16},\frac{1}{4},\frac{5}{24}\right)=\frac{\text{LCM}(5,1,5)}{\text{HCF}(16,4,24)}=\frac{5}{4}\text{hours}=1.25\text{hours}$

Thus, all three cyclists meet simultaneously at the starting point every 1.25 hours (or 1 hour 15 minutes).

Step 3: Calculate the total number of meetings.

The total continuous riding duration from 9:00 AM to 3:00 PM is 6 hours.

Listing the instances when the cyclists meet at the starting point during their ride:

1. At 0 hours (9:00 AM) — Start / 1st meeting

2. At 1.25 hours (10:15 AM) — 2nd meeting

3. At 2.50 hours (11:30 AM) — 3rd meeting

4. At 3.75 hours (12:45 PM) — 4th meeting

5. At 5.00 hours (2:00 PM) — 5th meeting

The next meeting would occur at 6.25 hours (3:15 PM), which falls outside the 6-hour period.

Therefore, counting all instances throughout the designated ride period up to 5 hours, all three cyclists meet simultaneously 5 times.