Problem 27 · 2015 Math Kangaroo
Stretch
Algebra & Patterns
work-backwardarithmetic-series
If one of the numbers 1, 2, 3, …, n − 1, n, is crossed out, the average of the remaining numbers is 4.75. Which number was crossed out?
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Answer: B — 7
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Hint 1 of 2
The remaining n−1 numbers average 4.75, so their total is 4.75(n−1); compare to 1+2+...+n.
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Hint 2 of 2
Try the n that makes 1+2+...+n minus a number in range equal 4.75(n−1); n = 9 works.
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Approach: match the sum after removal to the average
- For 1..n the full sum is n(n+1)/2; removing x leaves average 4.75 over n−1 numbers.
- Testing, n = 9 gives full sum 45; 4.75 × 8 = 38, so the removed number is 45 − 38 = 7.
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