Problem:
All the students in an algebra class took a 100-point test. Five students scored 100 , each student scored at least 60 , and the mean score was 76 . What is the smallest possible number of students in the class?
Answer Choices:
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Solution:
Each score of 100 is 24 points above the mean, so the five scores of 100 represent a total of points above the mean. Those scores must be balanced by scores totaling 120 points below the mean. Since no student scored more than points below the mean, the number of other students in the class must be an integer no less than 120/16. The smallest such integer is 8 , so the number of students in the class is at least 13 . Note that the conditions of the problem are met if 5 students score 100 and 8 score 61 .
OR
If there are students in the class, the sum of their scores is . If the five scores of 100 are excluded, the sum of the remaining scores is . Since each student scored at least 60 , the sum is at least . Thus
so . Since must be an integer, .
The problems on this page are the property of the MAA's American Mathematics Competitions