Let Angela's age = a
Let Bill's age = b
Let Charlie's age = c
a =2(b+c)
So, we want to find how old is Charlie (i.e the value of c).
Statement 1: Four years from now, the sum of all these people's ages will be 108.
(a+4) + (b+4) + (c+4) = 108
a+b+c+12 = 108
a+b+c = 108 - 12 = 96
where a = 2(b+c)
2b+2c+b+c = 96
3b + 3c = 96
(b+c) = 96/3 = 32
Here, the value of c is unknown; there, we cannot estimate the vale of c from this expression. Hence, statement 1 is NOT SUFFICIENT.

Statement 2: Five years ago, the average of all three people's age was 27.
Five years
$$\frac{\left(a-5\right)+\left(b-5\right)+\left(c-5\right)}{3}=27$$
a+b+c-15 = 81
a+b+c = 81+15 = 96
where a = 2(b+c)
2(b+c) + b + c = 96
3b + 3c = 96
b + c = 32
Also, 'c' is unknown. Hence, statement 2 is NOT SUFFICIENT.

Combining both statements together:
Both statements contain the same information (b+c = 32), which does not provide us with the value of c, hence, both statements combined together are NOT SUFFICIENT.

Answer = option E