2006 AMC 10B Problems/Problem 22
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 1
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 pairs 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
Solution 2
Note as above, you get the equation
Notice that we can multiply by on both sides to get whole numbers. Hence
Note that the prime factorization of .
Hence, we want or
Now, we have two cases to test.
Case 1:
Notice that we want or
Taking
Hence, .
Hence, the price of the jam is .
Solution 3 (Video)
Video solution: https://www.youtube.com/watch?v=7248rEcCSfM
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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