Problem 11 · 2012 AMC 8
Easy
Arithmetic & Operations
mean-median-mode
The mean, median, and unique mode of the positive integers 3, 4, 5, 6, 6, 7, and x are all equal. What is the value of x?
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Answer: D — 11.
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Hint 1 of 2
Three things must be equal, but one of them is already pinned down no matter what x is. Which one? The mode — 6 is the only value that already repeats, and x can't be allowed to create a tie.
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Hint 2 of 2
Once you know the common value, the rest is bookkeeping: mean = that value turns the whole thing into one equation, since mean fixes the total.
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Approach: pin the mode first — it forces the common value, then mean gives x
- Start with the most rigid clue. "Unique mode" means exactly one value repeats; only 6 does, so the mode is locked at 6 (and x must avoid 3, 4, 5, 7 or there'd be a tie). So all three equal 6.
- Now mean = 6 pins the total: 7 numbers averaging 6 must sum to 7 × 6 = 42.
- The six known values sum to 31, so x = 42 − 31 = 11.
- Check the median: sorted, the list is 3, 4, 5, 6, 6, 7, 11 — the middle (4th) value is 6 ✓. Lesson: in mean/median/mode puzzles, attack the most-constrained statistic first; it usually reveals the target number.
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