Problem 12 · 1996 AJHSME
Medium
Arithmetic & Operations
average-sum
What number should be removed from the list 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 so that the average of the remaining numbers is 6.1?
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Answer: B — 5.
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Hint 1 of 2
An average is just a disguised total: average × count = sum. The removed number is whatever's missing between the original total and the total the survivors must add up to.
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Hint 2 of 2
Get the sum of all 11 numbers, then the sum the remaining 10 need (6.1 × 10). The gap between those two sums IS the number you took out.
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Approach: turn averages back into sums
- Convert the target average to a total: the 10 remaining numbers must sum to 6.1 × 10 = 61. All eleven numbers (1 through 11) sum to 66.
- Removing a number drops the total from 66 to 61, so the number removed is 66 − 61 = 5.
- Why this transfers: almost every 'average changes when you add/remove an item' problem cracks open by rewriting each average as a total (average × count). Compare totals, not averages.
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