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 22:19, 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 ¢