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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:  