Problem 26 · 2013 Math Kangaroo
Stretch
Algebra & Patterns
Logic & Word Problems
substitutionwork-backward
In the finishing order of a cross-country race there are twice as many runners behind Alex as there are ahead of Daniel, and 1.5 times as many behind Daniel as ahead of Alex. Alex finished in 21st place. How many runners finished the race?
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Answer: B — 41
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Hint 1 of 2
Alex is 21st, so 20 runners finished before him; turn each clue into a count.
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Hint 2 of 2
Write the total as one unknown and use the two ratio clues to solve.
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Approach: set up counts around the two runners
- Before Alex there are 20 runners, so behind Daniel = 1.5 × 20 = 30.
- Let T be the total; behind Alex = T − 21 and before Daniel = T − 31.
- Behind Alex = 2 × (before Daniel): T − 21 = 2(T − 31).
- Solving: T − 21 = 2T − 62 → T = 41 runners.
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