A, B, and C can do a piece of work in 36 days, 54 days, and 108 days, respectively. They started the work together but A left 9 days after the start of the work and B left 3 days before the work was completed. For how many days did C work?

Let's assume the total work is 108 units.

A, B, and C can do a piece of work in 36 days, 54 days, and 108 days, respectively.

Efficiency of A = $$\frac{108}{36}$$ = 3 units/day

Efficiency of B = $$\frac{108}{54}$$ = 2 units/day

Efficiency of C = $$\frac{108}{108}$$ = 1 unit/day

They started the work together but A left 9 days after the start of the work and B left 3 days before the work was completed.

Let's assume the number of days C worked after A left is 'y'.

$$(3+2+1)\times9 + 2\times(y-3) + 1\times y = 108$$

$$6\times9+2y-6+y=108$$

$$54+3y-6=108$$

48+3y = 108

3y = 108-48 = 60

y = 20

The total number of days C work = (9+y)          [Here 9 is added. because they worked together for 9 days.]

= 9+20

= 29 days