Mr. X finishes 60% of a work in 9 days, and then is joined by Mr. Y and they together finish the next 10% work in 1 day. How long would it take if Y alone is asked to finish the next 5% of the work?

X can do 60% of the total work in 9 days, this gives us that X can complete the total work in 15 days. 
X and Y can together complete 10% of the work in 1 day, this gives us that X and Y can complete the work in 10 days. 

We can take the total work to be a multiple of the LCM of these values, let it be 30U. 
Now X has an efficiency of 2U units/day.
And X and Y together have an efficiency of 3U units/day. 

This gives us that the efficiency of Y to be 1U units/day.
In order for Y to 5% of the total work, that is 3/2 U units of work, Y would require, $$\frac{3}{2\ }=\ 1\ \frac{1}{2}$$ days. 

Hence, Option A is the correct answer.