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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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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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<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 ==

Revision as of 13:57, 25 October 2015

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

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

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