Problem 3 · 2009 Math Kangaroo
Medium
Algebra & Patterns
substitutionoff-by-one
2009 people are taking part in a public fun run. The number of people Hans beat is three times as big as the number of people who finished before him. In which place did Hans finish the race?
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Answer: A — 503rd
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Hint 1 of 2
Let b be the number who finished before Hans; then the number he beat is 3b.
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Hint 2 of 2
Everyone except Hans is either ahead of him or behind him — set up one equation.
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Approach: split the field into ahead, Hans, and behind
- Besides Hans there are 2008 runners: those ahead (b) and those he beat (3b).
- So b + 3b = 2008, giving 4b = 2008 and b = 502.
- Hans is one place behind those 502 runners, so he finishes in place 503.
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