You may find number of approaches for such questions. I personally find below method to be shortest.
A=36
B=24
C=72.

Take LCM of these numbers to get 18X12.
So A will finish 18X12/36=6 units in 1 day, similarly B=4 units, C=3 units.
Let total work to be completed is 18X12(This is imp number to choose)
Let C worked for x days
so A worked for x-8, B for x-12, C for x
so A's Rate X Days Worked + .... = total work
6(x-8)+4(x-12)+3x=18X12, solve for x to get 24.

I know this method works well, but I have a general question , is it OK to assume two values in one linear equation, here we assumed, 18X 12 and x, had it been ratio or similar, the assumed expression might have got canceled.Please comment.

In one day,

• A does 1/36 of the work
  B does 1/54 of the work
  C does 1/72 of the work
  
  Together all three of them does (1/36 + 1/54 + 1/72) = (1/18)*(1/2 + 1/3 + 1/4) = (1/18)*(6 + 4 + 3)/12 = 13/12*18

A and C together worked for (12 - 8) = 4 days.
In these 4 days, they completed 4*(1/36 + 1/72) = 4*(2/72 + 1/72) = 4*3/72 = 12/72 of the work

C alone worked for 8 days.
In these 8 days, C completed 8/72 of the work.

So, in last 12 days (12/72 + 8/72) = 20/72 of the work was done.
Hence, the rest of the work, i.e. (1 - 20/72) = 52/72 of the work was done by all of the together.

Now, all of the together takes 12*18/13 days to complete the job.
So, they will take (12*18/13)*(52/72) = 12 days to complete 52/72 of the work.

So, C worked for (12 + 12) = 24 days.