P alone can complete a piece of work in 10 days while Q alone can complete the same piece of work in 15 days. To complete the work in 4 days, R is hired by P and Q. How many days does R take to complete the work alone ?

12 days 

Efficiency ∝ $\frac{1}{time}$

                   P  :  Q

Time           10 : 15   = 2 : 3              

Efficiency      3 : 2             

Let the total work be 30

To complete this work in 4 days, R is hired with efficiency (say, r)

Thus, using the formula 

Efficiency × Days = Total work

 (3 + 2 + r) x 4   =  30

         20 + 4r      = 30

                   r      = 2.5

Time taken by R alone to complete the work = $\frac{Total\;work}{efficiency}$      = $\frac{30}{2.5}$

                                      = 12 days