Two runners run around a circular track. The first runner completes a lap in 4 minutes. The second runner completes a lap in 19 minutees. If they both

The faster rummer completes 1/4 of a lap in one minute when the slower runner completes 1/19  of a lap in one minute

So  

(1/4 - 1/19)T = 1

(15/76)T = 1    multiply both sides by  76/15

T = 76/15 min  = about 5.066 min

Proof

In 5.066min the faster runner runs  5.066(1/4)   = 1.266 laps

In 5.066min the slower runner runs 5.066(1/19)  = .266 laps

Thus....the first runner runs one more lap in 5.066 min

If they are to meet again at the starting place then we need a time that both 4 and 19 go into. So looking at multiples of 19 : 19, 38, 57, 76, 95.... 4 goes into 76. There fore the next time they meet at the starting place will be 76 minutes after they start