AMC12 2006 12B, Problem 14

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?

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 $1.65$ $\fbox{1.65}$.

Answer: 1.65