Difference between revisions of "2006 AMC 12B Problems/Problem 14"
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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> cents per glob and <math>J</math> blobs of jam at <math>5</math> cents per glob. 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 all positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches? | ||
+ | <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 == | ||
+ | From the given, we know that | ||
+ | |||
+ | <math>253=N(4B+5J)</math> | ||
+ | (The numbers are in cents) | ||
+ | |||
+ | since <math>253=11\cdot23</math>, and since <math>N</math> is an integer, then <math>4B+5J=11</math> or <math>23</math>. It is easily deduced that <math>11</math> is impossible to make with <math>B</math> and <math>J</math> integers, so <math>N=11</math> and <math>4B+5J=23</math>. Then, it can be guessed and checked quite simply that if <math>B=2</math> and <math>J=3</math>, then <math>4B+5J=4(2)+5(3)=23</math>. The problem asks for the total cost of jam, or <math>N(5J)=11(15)=165</math> cents, or <math>1.65\implies\mathrm{(D)}</math> | ||
== See also == | == See also == | ||
− | + | {{AMC12 box|year=2006|ab=B|num-b=13|num-a=15}} | |
+ | {{MAA Notice}} |
Latest revision as of 17:21, 8 December 2013
Problem
Elmo makes sandwiches for a fundraiser. For each sandwich he uses globs of peanut butter at cents per glob and blobs of jam at cents per glob. The cost of the peanut butter and jam to make all the sandwiches is . Assume that , and are all positive integers with . What is the cost of the jam Elmo uses to make the sandwiches?
Solution
From the given, we know that
(The numbers are in cents)
since , and since is an integer, then or . It is easily deduced that is impossible to make with and integers, so and . Then, it can be guessed and checked quite simply that if and , then . The problem asks for the total cost of jam, or cents, or
See also
2006 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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 12 Problems and Solutions |
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