2006 AMC 10B Problems/Problem 22
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[hide]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 1
The peanut butter and jam for each sandwich costs +5J\centNN(4B+5J)\cent$.
Setting this equal to$ (Error compiling LaTeX. Unknown error_msg)253\centN(4B+5J)=253=11\cdot23(N , 4B+5J)253 (1,253) ; (11,23) ; (23,11) ; (253,1) N>1BJ$.
So,$ (Error compiling LaTeX. Unknown error_msg)N=114B+5J=23BJB=2J=33\cdot5\cdot11=165\cent$$ (Error compiling LaTeX. Unknown error_msg)= $1.65 \Rightarrow D $
Solution 2
Note as above, you get the equation
Notice that we can multiply by on both sides to get whole numbers. Hence
Note that the prime factorization of .
Hence, we want or
Now, we have two cases to test.
Case 1:
Notice that we want or
Taking
Hence, .
Hence, the price of the Jam 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 |
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