KöMaL 2013 C1196

There are four children in a certain family. Santa brought them some jelly filled candies. The kids re-distributed the candies as follows: Olivia gave half of her candies to Peter. Then Peter generously passed on one third of all candies with him to Rob, and Rob passed on one fourth of his candies to Sarah. Then Sarah observed: ``If I gave Olivia one fifth of my candies, then each of us would have the same number of candies.'' What was the initial distribution of candies? What is the smallest possible number of all candies that the kids have?