Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2006 AMC 10B Problems/Problem 22"

m (added category)
(→‎Solution)
Line 9: Line 9:
 
Setting this equal to <math>253</math>¢:
 
Setting this equal to <math>253</math>¢:
  
−
<math>N(4B+5J)=253=11\cdot13</math>
+
<math>N(4B+5J)=253=11\cdot23</math>
  
 
The only possible positive integer pairs <math>(N , 4B+5J)</math> whose product is <math>253</math> are: <math> (1,253) ; (11,23) ; (23,11) ; (253,1) </math>
 
The only possible positive integer pairs <math>(N , 4B+5J)</math> whose product is <math>253</math> are: <math> (1,253) ; (11,23) ; (23,11) ; (253,1) </math>
Line 19: Line 19:
 
The only integer solutions for <math>B</math> and <math>J</math> are <math>B=2</math> and <math>J=3</math>
 
The only integer solutions for <math>B</math> and <math>J</math> are <math>B=2</math> and <math>J=3</math>
  
−
Therefore the cost of the jam Elmo uses to make the sandwiches is <math>3\cdot5\cdot11=165</math>¢ <math> = $1.65 \Rightarrow D </math>
+
Therefore the cost of the jam Elmo uses to make the sandwiches is <math>3\cdot5\cdot11=165</math>¢ <math> = </math>1.65 \Rightarrow D $
  
 
== See Also ==
 
== See Also ==

Revision as of 17:42, 11 May 2008

Problem

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 $N>1$. 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

The peanut butter and jam for each sandwich costs $4B+5J$¢, so the peanut butter and jam for $N$ sandwiches costs $N(4B+5J)$¢.

Setting this equal to $253$¢:

$N(4B+5J)=253=11\cdot23$

The only possible positive integer pairs $(N , 4B+5J)$ whose product is $253$ are: $(1,253) ; (11,23) ; (23,11) ; (253,1)$

The first pair violates $N>1$ and the third and fourth pair have no positive integer solutions for $B$ and $J$.

So, $N=11$ and $4B+5J=23$

The only integer solutions for $B$ and $J$ are $B=2$ and $J=3$

Therefore the cost of the jam Elmo uses to make the sandwiches is $3\cdot5\cdot11=165$¢ $=$1.65 \Rightarrow D $

See Also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2006_AMC_10B_Problems/Problem_22&oldid=25975"