Problem 9 · 2025 AMC 8
Medium
Arithmetic & Operations
arithmetic-series
Show answer
Answer: B — 6.5.
Show hints
Hint 1 of 2
You don't need to know which numbers are paired. The 6 pairs use up all 12 clock numbers, each exactly once — so what are you really averaging?
Still stuck? Show hint 2 →
Hint 2 of 2
Averaging the six pair-averages (each pair the same size) just re-averages all 12 numbers. So the answer is simply the average of 1 through 12.
Show solution
Approach: average of equal-size pair-averages = overall average
- Notice the actual pairings (1&2 across from 7&8, etc.) never matter: the six pairs cover all twelve numbers 1–12 exactly once. Averaging six equal-size pair-averages is the same as averaging all twelve numbers at once.
- And 1–12 are evenly spaced, so their average is just the midpoint of the ends: (1 + 12)/2 = 6.5.
- Why this transfers: averaging the averages of equal-size groups equals the overall average — but only when the groups are the same size. (Unequal groups need a weighted average.) And the mean of any evenly-spaced list is the midpoint of its first and last term.
Mark:
· log in to save