2011 AMC 12A Problems/Problem 7
Contents
Problem
A majority of the students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than . The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was . What was the cost of a pencil in cents?
Solution
The total cost of the pencils can be found by .
Since is the product of three sets of values, we can begin with prime factorization, since it gives some insight into the values: . Since neither nor are any of these factors, they can be eliminated immediately, leaving , , and .
Beginning with , we see that the number of pencils purchased by each student must be either or . However, the problem states that the price of each pencil must exceed the number of pencils purchased, so we can eliminate this.
Continuing with , we can conclude that the only case that fulfils the restrictions are that there are students who each purchased such pencils, so the answer is . We can apply the same logic to as we applied to if one wants to make doubly sure.
Video Solution
https://www.youtube.com/watch?v=6tlqpAcmbz4 ~Shreyas S
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 6 |
Followed by Problem 8 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.