A man takes 16 days to complete a project. A woman does the same work in 12 days. They started the work together on alternate days and the man worked on the first day. How much time will they take to complete the work?

$\frac{55}{4}$ days

Man = 16 days, Woman = 12 days,

⇒ They both work on alternate days, starting with Man;

 1st day = 3 units,

 2nd day = 4 units,

Every 2 days, & units of work is done,

So, in 12 days, 42 units is done.

and in 13 days, 45 units is done, 

Therefore, in \( {13 }_{ 4}^{3 } \) = \(\frac{55}{4}\) days, total work is done.