Let "a" be the rate of work of A;
    "b" ---------- ''--------- B:,  and
    "c" ---------- ''--------- C.


Then from the condition, you have these three equations for 3 unknowns


3a + 3b +  3c = 1,    (1)

4a      + 10c = 1,    (2)

     5b +  3c = 1.    (3)


The setup is done, which is the key step in the solution.


Now you can use any method to solve the system. For example, you can express "a" via "c" from (2) and express "b" via "c" from (3), 
and then substitute it into eq(1) to find "c".


As soon as you find "c", you will be able to find each "a" and "b" separately from eq(2) and eq(3) respectively.


Answer.  a = ,  b = ,  c = .


         It means that A needs 9 days;  B needs 6 days,  and C needs 18 days to complete the job, if everybody works alone.