2006 AMC 12B Problems/Problem 14

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$ (Error compiling LaTeX. Unknown error_msg)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 • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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2006 AMC 12B Problems/Problem 14

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