Bunuel wrote:
A certain experimental mathematics program was tried out in 2 classes in each of 32 elementary schools and involved 37 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1, but not more than 3, of the classes. If the number of teachers who taught 3 classes is n, then the least and greatest possible values of n, respectively, are
A) 0 and 13
B) 0 and 14
C) 1 and 10
D) 1 and 9
E) 2 and 8
This is a difficult problem, but if you break it up into small chunks, it begins to easily reveal itself.
1) What is the problem asking us to find? We need to find the (minimum, maximum) values for how many teachers could have taught 3 classes.
We are given the following: 64 total classes (2 classes per 32 elementary schools), 37 teachers, and the fact that each teacher can choose between 1, 2, or 3 classes.2) Looking at the answer choices, there is a much narrower range of potential minimums than maximums, so lets test minimums first to narrow down our answers. It is easier to test than to solve this algebraically.
We know there are 64 total classes. So let's add the first layer of classes. We know that every teacher teaches at least 1 class, so the total number of classes remaining for teachers after all teachers taught their first class is 64 - 37 = 27 classes (left for teachers' 2nd or 3rd class if they teach one).
We have 37 teachers and 27 classes left to cover, so it is very possible that 27 of the remaining 37 teachers choose to teach 2 classes, thus leaving no classes left. This gives us: Teachers with 1 class = 10, Teachers with 2 classes = 27, Teachers with 3 classes = 0.
We know that the minimum is 0. 3) Now, to find the maximum, let's say go back to our previous scenario after every teacher has taught their 1st class only. We have 27 classes remaining for teachers to choose a 2nd or 3rd class if they choose to do so. Let's say that 26 of the remaining classes are taught by 13 teachers, which means each teacher taught 3 classes. Since the max they can teach is 3 classes, a different teacher will need to teach the final 27th class remaining. This gives us: Teachers with 1 class = 23, Teachers with 2 classes = 1, Teachers with 3 classes = 13.
We know that the minimum must be 13. 4) Thus the correct answer is A. There is a minimum of 0 and a maximum of 13.