2009 AMC 12B Problem 1

Below is the professionally curated solution for Problem 1 of the 2009 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 12B solutions, or check the answer key.

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Concepts:parity money casework

Difficulty rating: 820

Each morning of her five-day workweek, Jane bought either a $50$ -cent muffin or a $75$ -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?

$1$

$2$

$3$

$4$

$5$

Solution:

With $b$ bagels she buys $5 - b$ muffins, costing $50(5-b) + 75b = 250 + 25b$ cents. For a whole number of dollars this must be a multiple of $100,$ so $25b$ must end in $50,$ meaning $b$ is even.

Among $b = 2$ and $b = 4,$ only $b = 2$ works: $250 + 50 = 300$ cents $= \$3.$

Thus, the correct answer is B.

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