A, B and C together can do a piece of work in 6 days. A and B together can do the same work in 12 days. B alone can do the same work in 18 days. In how many days can A and C together do the same work?

Let's assume the total work is 36 units.

A, B and C together can do a piece of work in 6 days.

The efficiency of A, B and C taken together = $$\frac{36}{6}$$ = 6 units/day    Eq.(i)

A and B together can do the same work in 12 days.

The efficiency of A and B taken together = $$\frac{36}{12}$$ = 3 units/day    Eq.(ii)

B alone can do the same work in 18 days.

The efficiency of B = $$\frac{36}{18}$$ = 2 units/day    Eq.(iii)

The efficiency of A = Eq.(ii)-Eq.(iii)

= (3-2) units/day

= 1 unit/day

The efficiency of C = Eq.(i)-Eq.(ii)

= (6-3) units/day

= 3 units/day

Time taken by A and C together do the same work = $$\frac{36}{1+3}$$

= $$\frac{36}{4}$$

= 9 days