Difference between revisions of "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 $\mathrm{(B)}$.

 2009 AMC 10B (Problems • Answer Key • Resources) Preceded byFirst Question Followed byProblem 2 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions
 2009 AMC 12B (Problems • Answer Key • Resources) Preceded byFirst Question Followed byProblem 2 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions