2025 AMC 8

Complete problem set with solutions and individual problem pages

Problem 9 Easy

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?

  • A.

    5

  • B.

    6.5

  • C.

    8

  • D.

    9.5

  • E.

    12

Answer:B

Solution 1

Our answer is

\frac{\frac{1+7}{2} + \frac{2+8}{2} + \cdots + \frac{6+12}{2}}{6} = \frac{\frac{1}{2}((1+7)+(2+8)+\cdots+(6+12))}{6} = \frac{1+2+3+4+5+6+7+8+9+10+11+12}{2 \cdot 6} = \frac{\frac{12 \cdot 13}{2}}{2 \cdot 6} = \frac{78}{12} =\boxed{\textbf{(B)}~6.5}

 

Solution 2

We proceed with brute force. All of the pairs of opposite numbers on the clock are (12,6), (1,7), (2,8), (3,9), (4,10), (5,11), where order doesn't matter. The averages of each of these pairs are 9, 4, 5, 6, 7, and 8 respectively, and the average these numbers is \frac{9+4+5+6+7+8}{6}=\boxed{\textbf{(B)}~6.5}

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