Q. 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.?

Explanation : 

1. Let t be the time in hours after 5:00 a.m.
2. Since P and Q walk in opposite directions, their relative speed (in rounds per hour) is
 5 + 3 = 8 rounds/hour.
3. They meet each time their combined distance equals an integer number of laps:
 8·t = k , where k = 1, 2, 3, …
so the k-th meeting happens at
 t = k/8 hours after 5:00.

4. We want all meetings between 5:20 a.m. and 7:00 a.m.
• 5:20 a.m. corresponds to t = 20/60 = 1/3 ≃ 0.3333 hours.
• 7:00 a.m. corresponds to t = 2 hours.

So we need
 1/3 ≤ k/8 ≤ 2
Multiply through by 8:
 8/3 ≤ k ≤ 16

• 8/3 ≃ 2.667 ⇒ the smallest integer k is 3.
• The largest k is 16.

5. Thus k = 3, 4, 5, …, 16. How many integers is that?
From 3 up to 16 inclusive there are 16 – 3 + 1 = 14 meetings.

Therefore, they cross each other 14 times between 5:20 and 7:00.