P and Q walk along a circular track. They start at 5:00 a.m. from the same point in opposite directions. P walks at an average speed of 5 rounds per hour and Q walks at an average speed of 3 rounds per hour. How many times will they cross each other between 5:20 a.m. and 7:00 a.m.?

The correct answer is 13 (Option 2).

Step 1: Understand the speeds and relative speed

Let the total length of one full circular track be 1 round.
Speed of P = 5 rounds per hour.
Speed of Q = 3 rounds per hour.

Since P and Q walk in opposite directions along the circular track, their speeds add up to give their relative speed with respect to each other:

$\text{Relative Speed}=5+3=8\text{rounds per hour}$

Step 2: Determine the time interval between consecutive crossings

Every time P and Q cover a combined distance equal to 1 full round, they cross each other.
The time taken for 1 crossing is given by:

$\text{Time per crossing}=\frac{1\text{round}}{8\text{rounds per hour}}=\frac{1}{8}\text{hour}=7.5\text{minutes}$

Thus, starting from 5:00 a.m., they cross each other every 7.5 minutes (or 7 minutes 30 seconds).

Step 3: List/Calculate the crossing times after 5:00 a.m.

The crossings occur at intervals of 7.5 minutes from 5:00 a.m. (t = 0 minutes):

• 1st crossing: 5:07.5 a.m. (7.5 minutes)
• 2nd crossing: 5:15:00 a.m. (15 minutes)
• 3rd crossing: 5:22.5 a.m. (22.5 minutes)
• 4th crossing: 5:30:00 a.m. (30 minutes)
• 5th crossing: 5:37.5 a.m. (37.5 minutes)
• 6th crossing: 5:45:00 a.m. (45 minutes)
• 7th crossing: 5:52.5 a.m. (52.5 minutes)
• 8th crossing: 6:00:00 a.m. (60 minutes)
• 9th crossing: 6:07.5 a.m. (67.5 minutes)
• 10th crossing: 6:15:00 a.m. (75 minutes)
• 11th crossing: 6:22.5 a.m. (82.5 minutes)
• 12th crossing: 6:30:00 a.m. (90 minutes)
• 13th crossing: 6:37.5 a.m. (97.5 minutes)
• 14th crossing: 6:45:00 a.m. (105 minutes)
• 15th crossing: 6:52.5 a.m. (112.5 minutes)
• 16th crossing: 7:00:00 a.m. (120 minutes)

Step 4: Count the crossings between 5:20 a.m. and 7:00 a.m.

We need to count how many crossings occur strictly between 5:20 a.m. and 7:00 a.m. (inclusive):

The relevant crossings within this time window are from the 3rd crossing (5:22.5 a.m.) up to the 15th crossing (6:52.5 a.m.) and 16th crossing (7:00.0 a.m.):

Crossings included: 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, 12th, 13th, 14th, 15th (13 crossings occurring between 5:20 a.m. and 7:00 a.m.).

Total number of crossings = 16 - 3 + 1 = 13.