A can finish a piece of work in 18 days and B can do the same work in half the time taken by A. Then working to their what part of the same work they can finish in a day?

Given that the amount of work completed by A alone is $\frac{1}{18}$

Since the time taken by B to complete the samee work is half of that of A

so that the amount of work completed by B alone is $\frac{2}{18}$

Hence the amount of work completed by both together is $\frac{1}{18}+\frac{2}{18}=\frac{3}{18}=\boxed{\frac{1}{6}}$