2006 AMC 12B Problems/Problem 14

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Problem

Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per glob. The cost of the peanut butter and jam to make all the sandwiches is $\$ $2.53$ . Assume that $B$ , $J$ and $N$ are all 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

From the given, we know that

$253=N(4B+5J)$ (The numbers are in cents)

since $253=11\cdot23$ , and since $N$ is an integer, then $4B+5J=11$ or $23$ . It is easily deduced that $11$ is impossible to make with $B$ and $J$ integers, so $N=11$ and $4B+5J=23$ . Then, it can be guessed and checked quite simply that if $B=2$ and $J=3$ , then $4B+5J=4(2)+5(3)=23$ . The problem asks for the total cost of jam, or $N(5J)=11(15)=165$ cents, or $1.65\implies\mathrm{(D)}$

2006 AMC 12B (Problems • Resources)
Preceded by Problem 13	Followed by Problem 15
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2006 AMC 12B Problems/Problem 14

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