Assume that Praetorian divides 3/5 pound cake into n pieces and 7/20 pound cake into m pieces. m and n pieces have equal weight
weight of each cake piece = (3/5)/n = 3/5n Since each cake piece has same weight
(3/5)/n = (7/20)/m
3/n = 7/4m therefore m = n * 7/12
Now we want minimum pieces because minimum pieces yeild maximum size and we know that m and n are integers
if n = 12 then m = 7
there fore praetorian will invite 7+12 = 19 guests.
I hope I am included