Hello.

Let's see your question.

Let's call the numbers a, b, c, d and e.

(1) The average of the numbers is zero

This implies that: $$\frac{a+b+c+d+e}{5}=0\ \ \ \Rightarrow\ \ a+b+c+d+e=0.$$ Hence, the sum of the five numbers is zero. SUFFICIENT.

(2) Only one of the numbers is positive.

This is not sufficient. We can have for example: a=b=c=d=0 and e=1, hence the sum is 1; but we also can have a=b=c=d=0 and e=2, and in this case the sum is 2. NOT SUFFICIENT.

The correct answer is the option A.