2006 AMC 10B Problems/Problem 22

Problem

Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ ¢ per glob and $J$ blobs of jam at $5$ ¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is $$2.53$ . Assume that $B$ , $J$ , and $N$ are positive integers with $N>1$ . What is the cost of the jam Elmo uses to make the sandwiches?

$\mathrm{(A) \ } $1.05\qquad \mathrm{(B) \ } $1.25\qquad \mathrm{(C) \ } $1.45\qquad \mathrm{(D) \ } $1.65\qquad \mathrm{(E) \ } $1.85$

Solution

The peanut butter and jam for each sandwich costs $4B+5J$ ¢, so the peanut butter and jam for $N$ sandwiches costs $N(4B+5J)$ ¢.

Setting this equal to $253$ ¢:

$N(4B+5J)=253=11\cdot13$

The only possible positive integer pairs $(N , 4B+5J)$ whose product is $253$ are: $(1,253) ; (11,23) ; (23,11) ; (253,1)$

The first pair violates $N>1$ and the third and fourth pair have no positive integer solutions for $B$ and $J$ .

So, $N=11$ and $4B+5J=23$

The only integer solutions for $B$ and $J$ are $B=2$ and $J=3$

Therefore the cost of the jam Elmo uses to make the sandwiches is $3\cdot5\cdot11=165$ ¢ $= $1.65 \Rightarrow D$ (Error compiling LaTeX. Unknown error_msg)

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2006 AMC 10B Problems/Problem 22

Problem

Solution

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