137

[vaivish]

137. If 37 teachers are to be assigned to 64 classes in such a way that each of teacher teaches at least one class and at most three classes. What are the greatest possible number and the least possible number of the teachers who teach three classes?
(A) 14,0
(B) 13, 1
(C) 13, 0
(D) 12, 2
(E) 12, 1


OA is c.

[kuroneko1313]

137. If 37 teachers are to be assigned to 64 classes in such a way that each of teacher teaches at least one class and at most three classes. What are the greatest possible number and the least possible number of the teachers who teach three classes?
(A) 14,0
(B) 13, 1
(C) 13, 0
(D) 12, 2
(E) 12, 1


OA is c.

At least the teachers must teach one class. The number of unassigned classes after fulfilling this requirement is 64 - 37 = 27.

For the maximum number of teachers who teach 3 classes:
27/2 = 13 with remainder of 1.
This means that there's going to be one teacher who teaches two classes.

For the minimum number of teachers who teach 3 classes:
Out of those 27 classes we can assign it evenly to the 37 teachers, so the minimum number of teachers who teach 3 classes is zero. There are going to be 10 teachers who only teach 1 class in this case.

[jeffxujian]

IMO C

I used back-substition as follows
put 14 in we have covered 42 classes with 23 teachers and 22 classes left.however, every teacher is required to teach at least one, so A is wrong.

I put B in, 39 classes covered with 25 classes and 24 teacher left Remember some teacher can teach 2 classes. so 13 satisfies the biggest possible number and we can, therefore, confidently eliminate D, E.

Now we only have B & C left, if you looking for the least possible#, start with the lowest number first. In thise case, it is okay for 0 people teach the maximum 3 classes. i.e 0*3+10*1+27*2=64