2025 AMC 8 Problems/Problem 9

Problem

Ningli looks at the 6 pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting 6 numbers?

$\textbf{(A)}\ 5\qquad \textbf{(B)}\ 6.5\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9.5 \qquad \textbf{(E)}\ 12$

Solution 1

Notice that this is simply the sum of the numbers on the clock face divided by $12$ . Why? This is because for a given number, it will be divided by $2$ due to its averaging with its counterpart on the other side of the clock face, and then it will be divided by $6$ when we average out all the smaller averages. Since the sum of the first $12$ positive integers is $78$ , our answer is $\frac{78}{12}=\boxed{\textbf{(B)}~6.5}$ . ~cxsmi

Solution 2

The pairs for the opposite numbers on the clock are $(12,6)$ , $(1,7)$ , $(2,8)$ , $(3,9)$ , $(4,10)$ , and $(5,11)$ . The averages of each of these pairs are $9, 4, 5, 6, 7,$ and $8$ respectively. The averages of $9, 4, 5, 6, 7,$ and $8$ are $\frac{39}{6}=\boxed{\textbf{(B)}~6.5}$