Question

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 $N \gt 1.$ What is the cost of the jam Elmo uses to make the sandwiches?

$\$1.05$

$\$1.25$

$\$1.45$

$\$1.65$

$\$1.85$

Answer: D

Concepts:Diophantine Equation prime factorization casework

Difficulty rating: 1680

Solution:

The total cost in cents is $N(4B + 5J) = 253 = 11 \cdot 23.$ Since $N \gt 1,$ the value of $N$ is $11,$ $23,$ or $253.$

If $N = 253$ then $4B + 5J = 1,$ and if $N = 23$ then $4B + 5J = 11;$ neither has a positive integer solution.

So $N = 11$ and $4B + 5J = 23,$ whose only positive solution is $B = 2,$ $J = 3.$

The jam costs $N \cdot J \cdot 5$ ¢ $= 11 \cdot 3 \cdot 5 = 165$ cents, or $\$1.65.$

Thus, the correct answer is D.

2006 AMC 12B Problem 14

Answer: D

Difficulty rating: 1680

Solution:

Problem 14 in Other Years