$A$ alone can complete a work in $16$ days and $B$ alone in $12$ days. Starting with $A,$ they work on alternate days. The total work will be completed in :

The correct option is D $13\frac{3}{4}$ days
On the first day A complete; $\frac{1}{16}th$ of work.

On the second day B complete; $\frac{1}{12}th$ of work.

Therefore in $2$ days $=\frac{1}{16}+\frac{1}{12}=\frac{7}{48}$ of the work.

In $12$ days they complete $6\times \frac{7}{48}$ of the work $=\frac{42}{48}$ of the work

Remaining work $=\frac{6}{48}$

On the 13th day A completes $\frac{3}{48}$ of the work.

Remaining work is $\frac{3}{48}$ now B completes work in $\left(\frac{3}{48}\right)\left(\frac{1}{12}\right)=\frac{3}{4}$ day.

Whole work completed in $13\frac{3}{4}$days.

Hence, the answer is $13\frac{3}{4}.$