Problem 2 · 2014 AMC 8
Easy
Arithmetic & Operations
extremes-min-max
Paul owes Paula 35 cents and has a pocket full of 5-cent coins, 10-cent coins, and 25-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
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Answer: E — 5 coins difference.
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Hint 1 of 2
"Most coins" and "fewest coins" are opposite greedy goals for the same total: to pile up the most, use the smallest coin every time; to use the fewest, grab the biggest coin that still fits.
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Hint 2 of 2
Find each count separately, then subtract.
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Approach: two opposite greedy choices
- Most coins ⇒ use only the smallest coin (5 cents): 35 ÷ 5 = 7 coins.
- Fewest coins ⇒ grab the largest coins first: 25 + 10 = 35 uses just 2 coins.
- Difference: 7 − 2 = 5.
- You'll see this again: "maximize the count" and "minimize the count" almost always mean "use the smallest pieces" vs. "use the largest pieces" — a greedy idea that works for making change, stamps, and weights.
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