2004 AMC 12B Problems/Problem 11
Problem
All the students in an algebra class took a -point test. Five students scored , each student scored at least , and the mean score was . What is the smallest possible number of students in the class?
Solution
Let the number of students be . Then the sum of their scores is at least . At the same time, we need to achieve the mean , which is equivalent to achieving the sum .
Hence we get a necessary condition on : we must have . This can be simplified to . The smallest integer for which this is true is .
To finish our solution, we now need to find one way how students could have scored on the test. We have points to divide among them. The five s make , hence we must divide the remaining points among the other students. This can be done e.g. by giving points to each of them.
Hence the smallest possible number of students is .
See Also
2004 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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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