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

 
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== Problem ==
 
== Problem ==
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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?
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<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>
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== Solution ==
 
== Solution ==
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The peanut butter and jam for each sandwich costs <math>4B+5J</math>¢, so the peanut butter and jam for <math>N</math> sandwiches costs <math>N(4B+5J)</math>¢.
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Setting this equal to <math>253</math>¢:
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<math>N(4B+5J)=253=11\cdot13</math>
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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>
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The first pair violates <math>N>1</math> and the third and fourth pair have no positive integer solutions for <math>B</math> and <math>J</math>.
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So, <math>N=11</math> and <math>4B+5J=23</math>
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The only integer solutions for <math>B</math> and <math>J</math> are <math>B=2</math> and <math>J=3</math>
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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>
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== See Also ==
 
== See Also ==
 
*[[2006 AMC 10B Problems]]
 
*[[2006 AMC 10B Problems]]

Revision as of 16:45, 16 July 2006

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 $253$¢:

$N(4B+5J)=253=11\cdot13$

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 $3\cdot5\cdot11=165$¢ $= $1.65 \Rightarrow D$ (Error compiling LaTeX. Unknown error_msg)

See Also

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