Problem 15 · 2013 Math Kangaroo
Medium
Fractions, Decimals & Percents
Number Theory
casework
Willi wrote down a few consecutive whole numbers. A certain percentage of them are odd. Which of the following values cannot be that percentage?
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Answer: B — 45%
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Hint 1 of 2
In any run of consecutive integers the count of odd numbers is about half.
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Hint 2 of 2
Write the percentage as (odd count)/(total) and test which fraction is impossible.
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Approach: odd-count fraction must be achievable
- Among n consecutive whole numbers, the number of odd ones is n/2 (rounded up or down).
- So the percentage is (odd count)/n, which can be 40, 48, 50, 60 for suitable n.
- 45% = 9/20 would need 9 odds out of 20 consecutive numbers, but 20 consecutive numbers always contain exactly 10 odds.
- So 45% cannot occur.
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