Latest figures
FG has posted a solution for the 11x1 rectangle, attaining 101/27 = 3.740740...
More results:
7x3 rectangle is 56/15 = 3.733333...
7x2 rectangle is 67/18 = 3.722222...
8x1 rectangle is 26/7 = 3.714285...
6x1 rectangle is 85/23 = 3.695652...
7x1 rectangle is 11/3 = 3.666666...
5x1 rectangle is 25/7 = 3.571428...
I have removed specific solutions from this answer, as they have been superseded by FG's results.