AbdurRakib wrote:
If x, y, and z are positive numbers, what is the value of the average (arithmetic mean) of x and z ?
(1) x - y = y - z
(2) x^2 - y^2 = z
We need to determine (x + z)/2.
Statement One Alone:
x - y = y - z
Simplifying the equation, we have:
x - y = y - z
x + z = 2y
Since x + z = 2y, we have:
(x + z)/2 = 2y/2 = y
However, since we do not know the value of y, we cannot determine the average. Statement one alone is not sufficient to answer the question.
Statement Two Alone:
x^2 - y^2 = z
This does not provide enough information to determine the average of x and z. For example, if x = 3 and y = 2, then z = 5, and the average of x and z would be 4. However, if x = 4 and y = 2, then z = 12, and the average of x and z would be 8. Statement two alone is not sufficient to answer the question.
Statements One and Two Together:
From statement one, we see that z = 2y - x, and from statement two, we have z = x^2 - y^2. Thus, we have 2y - x = x^2 - y^2.
Notice that the equation above has two variables; thus, there are infinitely many solutions. That is, we won’t have a unique value for x or y, and hence we don’t have a unique value for z, either. The two statements together are still not sufficient to answer the question.
Answer: E