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

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 ¢ $= $1.65 \Rightarrow D$ (Error compiling LaTeX. Unknown error_msg)

See Also

• 2006 AMC 10B Problems

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