1973 AHSME Problem 5

Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean),

\text{I. Averaging is associative }

\text{II. Averaging is commutative }

\text{III. Averaging distributes over addition }

\text{IV. Addition distributes over averaging }

\text{V. Averaging has an identity element }

those which are always true are

\textbf{(A)}\ \text{All}\qquad\textbf{(B)}\ \text{I and II only}\qquad\textbf{(C)}\ \text{II and III only}

\textbf{(D)}\ \text{II and IV only}\qquad\textbf{(E)}\ \text{II and V only}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.