P and Q working separately can do a piece of work in 9 and 12 days, respectively. Beginning with P, they work alternately for a day. In how many days will the work be completed?

Let us say the work is 36 units big. 
This means that P, who can finish the work in 9 days, does 4 units a day, and Q, who can finish the work in 12 days, does 3 units a day. 

Working alternatively means that every two days (4+3) units of work will be done. 
So, 5 cycles of this will ensure 35 units of work is completed, 5 cycles means 10 units of work. 
The last unit is completed by the person who starts work first, which is P. P does 4 units a day so that P can finish 1 unit in 0.25 days. 
Hence, the answer will be 10.25 days.