Question

A and b start together from the same point on a circular track and walk in the same direction till they both again arrive together at the starting point. a completes one circle in 224 s and b in 364 s. how many times will a have passed b?

Answer 1

For A and B to arrive together at the starting point,
each of them must have completed an integer number of turns.
The time (in seconds) required for that must be a multiple of  224 and of 364.

224 = 2 raise to power 5.7 and 364 = 2 raise to power 2 . 7. 13

so the minimum common multiple of  224 and 364  is 2912 = 2 raise to power 5 . 7. 13

That means that A and B first arrive together at the starting point  2912 seconds after the start. .
During that time, A will have run 2912/224 = 13  laps, and B will have run 2912/364 = 8 laps

Each time the distance A has run exceeds the distance B has run by an integer (1, 2, 3, or 4 laps), A is passing B.

When both have run for  2912 seconds, A has run 13-8 = 5 more laps than B, and they are both arriving together at the starting point.

so A passes B 41 times.