First arrange them in ascending order :

10.2 ,14.2, 15.6, 18.6, 19.8

Now, mean = \frac{78.4}{5}=15.68

and median = 15.6

Now, we need to add minimum in such a way that either of the following condition are satisfied :

- Mean \geq 16.5
- Mean > 16 \text{ and} \ Median\geq16.5

Now, eliminate option A immediately as we need a value greater than our current mean(i.e. 15.68) to drag the value towards 16.

If we treat it like a weighted average problem we , can quickly solve it:

Now, tackle mean \geq 6 first ;

10.2 is 5.8 unit away from 16

14.2 is 1.8 unit away from 16

15.6 is .4 unit away from 16

18.6 is 2.6 ahead of 16

19.8 is 3.8 ahead of 16

or the above calculation can be re-written as:

-5.8 + -1.8 + -0.4 + 2.6 + 3.8

if we sum it , we’ll end up with -1.6 , thus we need our answer to be atleast 1.6 unit ahead of 16 (which is option C 17.6 ) as 17.6 -16 = 1.6

now checking on median condition :

10.2 ,14.2, (15.6 , 17.6) , 18.6, 19.8

median = \frac{15.6 + 17.6}{2}=16.6

Thus, option C is the min value which meets our condition