Difference between revisions of "2006 AMC 10B Problems/Problem 22"
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The only integer solutions for <math>B</math> and <math>J</math> are <math>B=2</math> and <math>J=3</math> | The only integer solutions for <math>B</math> and <math>J</math> are <math>B=2</math> and <math>J=3</math> | ||
| − | Therefore the cost of the jam Elmo uses to make the sandwiches is <math>3\cdot5\cdot11=165</math>¢ <math> = </math>1.65 \Rightarrow D | + | Therefore the cost of the jam Elmo uses to make the sandwiches is <math>3\cdot5\cdot11=165</math>¢ <math> = </math> <math>1.65 \Rightarrow D </math> |
== See Also == | == See Also == | ||
Revision as of 21:19, 7 January 2011
Problem
Elmo makes 
sandwiches for a fundraiser. For each sandwich he uses 
globs of peanut butter at 
¢ per glob and 
blobs of jam at 
¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is 
. Assume that , , and are positive integers with 
. What is the cost of the jam Elmo uses to make the sandwiches?
Solution
The peanut butter and jam for each sandwich costs 
¢, so the peanut butter and jam for sandwiches costs 
¢.
Setting this equal to 
¢:
The only possible positive integer pairs 
whose product is are: 
The first pair violates and the third and fourth pair have no positive integer solutions for and .
So, 
and 
The only integer solutions for and are 
and 
Therefore the cost of the jam Elmo uses to make the sandwiches is 
¢ 
