Question

Let $@$ denote the "averaged with" operation: $a @ b = \frac{a+b}{2}.$ Which of the following distributive laws hold for all numbers $x, y,$ and $z?$

I. $x @ (y + z) = (x @ y) + (x @ z)$

II. $x + (y @ z) = (x + y) @ (x + z)$

III. $x @ (y @ z) = (x @ y) @ (x @ z)$

I only

II only

III only

I and III only

II and III only

In statement I, the left-hand side equals $\dfrac{x+y+z}2$ and the right-hand side equals $x+\dfrac{y+z}2,$ so they are not always equal.

In statement II, both sides equal $x+\dfrac{y+z}2,$ so it holds.

In statement III, both sides equal $\dfrac{2x+y+z}4,$ so it also holds.

Thus, the correct answer is E .