Problem 7 · AMC 8 Stretch
Core
Logic & Word Problems
Arithmetic & OperationsCounting & Probability
logical-reasoningconsidering-extreme-cases
You win a lottery! There are three piles of bills: a 100-dollar pile, a 50-dollar pile, and a 10-dollar pile. You may take 10 bills from one pile, 5 bills from another, and 1 bill from the third (you choose which pile gets which count). Matching the counts to win the most money, how many dollars do you win?
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Answer: 1260 dollars
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Hint 1 of 4
This is like the dentist problem flipped: now you want the total to be as BIG as possible.
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Hint 2 of 4
Each pile's bill value gets multiplied by one of the counts 10, 5, or 1. Which count do you want on the most valuable bill?
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Hint 3 of 4
Put the biggest count on the biggest bill: take 10 bills of 100 dollars, 5 bills of 50 dollars, 1 bill of 10 dollars.
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Approach: Pair the largest multiplier with the largest value
- Your winnings are (some count) times each pile's value, added up. To make that largest, give the largest count to the largest bill.
- Take 10 bills from the 100-dollar pile, 5 from the 50-dollar pile, and 1 from the 10-dollar pile.
- \(10 \times 100 + 5 \times 50 + 1 \times 10 = 1000 + 250 + 10 = 1260\) dollars.
- So you win 1260 dollars — the mirror image of the dentist problem, where you matched the big multiplier to the small number instead.
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