2006 AMC 12B Problem 14

Problem 14 of 25IntermediateNumber Theory

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?

Show solution

Solution

The total cost in cents is

N(4B + 5J) = 253 = 11 \cdot 23.

Since

N \gt 1,

the value of

N

11,

23,

253.

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.