The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

In our School on the Range we have a grading system that goes from *1* to *50*. All the grades below *21* are failing grades while the grades from *21* inclusive to *50* are passing grades. There are *800* students in total. If each passing grade has been assigned to at least one student, there are no two passing grades assigned to the same number of students and the maximum number of students with the same grade is *38* what is the difference between the largest possible number of failing grades and the smallest possible number of failing grades?

For solving this problem we will use the sum of an arithmetic sequence.

From *21* to *50* there are *30* numbers. The largest possible number of passing grades happens when each grade is assigned the largest possible number of times. Since there are *30* passing grades and the largest number of students with the same grade is *38* then:

is the largest possible number of passing grades. Consequently,

is the smallest possible number of failing grades.

Similarly, the smallest possible number of passing grades can be found by allowing the smallest possible number of students in each grade category:

From here we find the largest possible number of failing grades as the difference:

The answer to the question asked is: