# 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>\$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 == | ||

+ | 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>¢. | ||

+ | |||

+ | Setting this equal to <math>253</math>¢: | ||

+ | |||

+ | <math>N(4B+5J)=253=11\cdot13</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> | ||

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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>. | ||

+ | |||

+ | 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 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?

## 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 ¢ $= $1.65 \Rightarrow D$ (Error compiling LaTeX. ! Missing $ inserted.)