Difference between revisions of "2006 AMC 10B Problems/Problem 22"
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== Problem == | == Problem == | ||
− | Elmo makes <math>N</math> sandwiches for a fundraiser. For each sandwich he uses <math>B</math> globs of peanut butter at <math>4</math>¢ per glob and <math>J</math> blobs of jam at <math>5</math>¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is <math> | + | Elmo makes <math>N</math> sandwiches for a fundraiser. For each sandwich he uses <math>B</math> globs of peanut butter at <math>4</math>¢ per glob and <math>J</math> blobs of jam at <math>5</math>¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is <math>$2.53</math>. Assume that <math>B</math>, <math>J</math>, and <math>N</math> are positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches? |
− | <math> \mathrm{(A) \ } | + | <math> \mathrm{(A) \ } $1.05\qquad \mathrm{(B) \ } $1.25\qquad \mathrm{(C) \ } $1.45\qquad \mathrm{(D) \ } $1.65\qquad \mathrm{(E) \ } $1.85 </math> |
== Solution == | == Solution == | ||
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[[Category:Introductory Number Theory Problems]] | [[Category:Introductory Number Theory Problems]] | ||
+ | {{MAA Notice}} |
Revision as of 12:18, 4 July 2013
Problem
Elmo makes sandwiches for a fundraiser. For each sandwich he uses
globs of peanut butter at
¢ per glob and
blobs of jam at
¢ per blob. The cost of the peanut butter and jam to make all the sandwiches is
. Assume that
,
, and
are positive integers with
. 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) \ } $1.85$ (Error compiling LaTeX. Unknown error_msg)
Solution
The peanut butter and jam for each sandwich costs ¢, so the peanut butter and jam for
sandwiches costs
¢.
Setting this equal to ¢:
The only possible positive integer pairs whose product is
are:
The first pair violates and the third and fourth pair have no positive integer solutions for
and
.
So, and
The only integer solutions for and
are
and
Therefore the cost of the jam Elmo uses to make the sandwiches is ¢
See Also
2006 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.