Problem 14 · 1997 AJHSME
Hard
Arithmetic & Operations
mean-median-mode
A set of five positive integers has mean 5, median 5, and 8 as its only mode. What is the difference between the largest and smallest integers in the set?
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Answer: D — 7.
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Hint 1 of 2
Each statistic is a clue that PINS DOWN actual numbers. Sort the five into slots ⬚ ⬚ ⬚ ⬚ ⬚ and let mean, median, mode each fill in what they force.
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Hint 2 of 2
Convert mean → a fixed total (sum = mean × count); convert 'only mode 8' → at least two 8s; convert median → the middle slot. Then the rest is forced.
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Approach: translate each statistic into a fixed slot, then solve for the rest
- Write the sorted set as a ≤ b ≤ 5 ≤ d ≤ e (median 5 fixes the middle). Mean 5 means the total is 5 × 5 = 25.
- 'Only mode is 8' forces two 8s, and they must be the two largest: d = e = 8 (16 used). So a + b = 25 − 16 − 5 = 4.
- a and b are distinct positive integers summing to 4 (distinct so 8 stays the *only* mode), forcing 1 and 3 → set {1, 3, 5, 8, 8}.
- Largest − smallest = 8 − 1 = 7.
- You'll see it again: with mean/median/mode puzzles, lay out sorted slots and spend your strongest constraints (total and mode) first — the leftover slots then have only one legal filling.
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