*** UPDATE *** My answer above is incorrect....
1/(A+B) = 6 1/(B+C) = 9 1/(C+A) = 12
B = (1-6A)/6 C = (1-12A)/12
1/ [ (1-6A)/6 + (1-12A)/12]
Solve for A = 15/216
then B= 21/216
and C = 4/216
1/(A+B+C) = x
1/(40/216) = x
216/40 = x = 5.4 days (IF I did all of the calcs w/out error!!)