Difference between revisions of "2009 AMC 10B Problems/Problem 1"

(Solution 2)
(Solution 2)
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== Solution 2 ==
 
== Solution 2 ==
Because <math>75</math> ends in a <math>5</math>, 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 <math>50</math> cents multiplied by an odd number means that it will end in a <math>50</math> , we know that the result of the even number multiplied by <math>75</math>,must end in a <math>50</math>. Note that the only result that gives this result is when 75 is multiplied by 2. Thus, our answer is <math>\mathrm{(B)}</math>.
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Because <math>75</math> ends in a <math>5</math>, 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 <math>50</math> cents multiplied by an odd number means that it will end in a <math>50</math> , we know that the result of the even number multiplied by <math>75</math> , must end in a <math>50</math>. Note that the only result that gives this result is when <math>75</math> is multiplied by <math>2</math>. Thus, our answer is <math>\mathrm{(B)}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 21:01, 27 April 2020

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 $\mathrm{(B)}$.

See also

2009 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 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 (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 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

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