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

Latest revision as of 09:55, 24 June 2024

Problem

Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4\cent$ per glob and $J$ blobs of jam at $5\cent$ 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 1

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

Setting this equal to $253\cent$ :

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

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 pairs 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 $3\cdot5\cdot11=165\cent$ $= $1.65 \Rightarrow \boxed{\mathrm{D}}$

Solution 2

Note as above, you get the equation $N(0.04B+0.05J)=2.53$

Notice that we can multiply by $100$ on both sides to get whole numbers. Hence $\implies N(4B+5J)=253$

Note that the prime factorization of $253=11\cdot23$ .

Hence, we want $4B+5J=23$ or $11$

Now, we have two cases to test.

Case 1: $4B+5J=23$

Notice that we want $B\le5$ or $J\le4$

Taking $\pmod{5}\implies 4B\equiv3\pmod{5}\implies B=2$

Hence, $B=2,J=3$ .

Hence, the price of the jam is $3\cdot11\cdot{0.05}\implies 1.65 \implies\boxed{D}$ .

Video Solutions

Video solution: https://www.youtube.com/watch?v=7248rEcCSfM

https://www.youtube.com/watch?v=dIaBGFwwZEc ~David

@@ 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>.
-Therefore, the cost of the jam Elmo uses to make the sandwiches is <math>3\cdot5\cdot11=165\cent</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\cent</math>  <math>=  \$1.65 \Rightarrow \boxed{\mathrm{D}}</math>
 ==Solution 2==
-Note as above, you get the equation <math>N(0.04B+0.05J=2.53)</math>
+Note as above, you get the equation <math>N(0.04B+0.05J)=2.53</math>
 Notice that we can multiply by <math>100</math> on both sides to get whole numbers. Hence <math>\implies N(4B+5J)=253</math>
@@ Line 41: / Line 41: @@
 Hence, <math>B=2,J=3</math>.
-Hence, the price of the Jam is <math>3\cdot11\cdot{0.05}\implies 1.65 \implies\boxed{D}</math>.
+Hence, the price of the jam is <math>3\cdot11\cdot{0.05}\implies 1.65 \implies\boxed{D}</math>.
-== Solution 3 (Video) ==
+== Video Solutions ==
 Video solution: https://www.youtube.com/watch?v=7248rEcCSfM
+https://www.youtube.com/watch?v=dIaBGFwwZEc   ~David
 == 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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