2009 AMC 12A Problem 4

Below is the professionally curated solution for Problem 4 of the 2009 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12A solutions, or check the answer key.

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Concepts:money multiple casework

Difficulty rating: 1200

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?

$15$

$25$

$35$

$45$

$55$

Solution:

If the four coins include a penny, the total is not a multiple of $5,$ so it cannot equal any of the five listed multiples of $5.$ If there is no penny, every coin is worth at least $5$ cents, so the total is at least $20$ cents. Either way, $15$ is impossible.

The other amounts are attainable: $25 = 10 + 5 + 5 + 5,$ $35 = 10 + 10 + 10 + 5,$ $45 = 25 + 10 + 5 + 5,$ and $55 = 25 + 10 + 10 + 10.$

Thus, the correct answer is A.

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