# 2009 AMC 10B Problems/Problem 1

The following problem is from both the 2009 AMC 10B #1 and 2009 AMC 12B #1, so both problems redirect to this page.

## Problem

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? $\mathrm{(A)}\ 1\qquad \mathrm{(B)}\ 2\qquad \mathrm{(C)}\ 3\qquad \mathrm{(D)}\ 4\qquad \mathrm{(E)}\ 5$

## Solution 1

The only combination of five items with total cost a whole number of dollars is 3 muffins and $\boxed {2}$ bagels. The answer is $\mathrm{(B)}$.

## Solution 2

Because $75$ ends in a $5$, and we want a whole number of dollars, we know that there must be an even number of bagels. Furthermore, this tells us that the number of muffins is odd. Now, because it is a whole number of dollars, and $50$ cents multiplied by an odd number means that it will end in a $50$ , we know that the result of the even number multiplied by $75$ , must end in a $50$. Note that the only result that gives this result is when $75$ is multiplied by $2$. Thus, our answer is $\mathrm{(B)}$.

~coolmath2017

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