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2024 Math Kangaroo

Problem 14

Problem 14 · 2024 Math Kangaroo Hard
Logic & Word Problems work-backwardoff-by-one

Andrew throws arrows at a target. He starts with 10 arrows. Each time he hits the target, he gets 2 more arrows. In total Andrew throws 20 arrows, and then he has run out of arrows. How many times did Andrew hit the target?

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Answer: B — 5
Show hints
Hint 1 of 3
He only started with 10 arrows, but he threw 20, so where did the extra arrows come from?
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Hint 2 of 3
Every hit is like a little gift of 2 more arrows.
Still stuck? Show hint 3 →
Hint 3 of 3
Figure out how many extra arrows he earned, then see how many hits that took.
Show solution
Approach: count the extra arrows the hits gave him
  1. He started with 10 arrows but threw 20 in total, so 10 extra arrows must have come from hitting the target.
  2. Each hit gives 2 extra arrows, so we count by twos: 2, 4, 6, 8, 10 — that takes 5 hits to reach 10 extra arrows.
  3. So Andrew hit the target 5 times.
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