Question: 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

“A certain experimental mathematics program was tried out in 2 classes”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide 2021”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:
Approach Solution 1:
x --> Number of teachers who teach in 3 subjects
y --> Number of teachers who teach in 2 subjects
z --> Number of teachers who teach in 1 subject

Total number of classes = 2 * 32 = 64

x + y + z = 37
3x + 2y + z = 64

Check the possibilities from the answer choices:
Put x = 0 --> y + z = 37 and 2y + z = 64 --> y = 27 and z = 10 (Possible minimum)
Put x = 14 --> y + z = 23 and 2y + z = 22 --> y = -1 (Not possible)
Put x = 13 --> y + z = 24 and 2y + z = 25 --> y = 1 and z = 23 (Possible maximum)

Correct Answer: A

Approach Solution 2:
We are given that a certain experimental mathematics program was tried out in 2 classes in each of 32 elementary schools and involved 37 teachers. Thus, there were a total of 2 x 32 = 64 classes under this program.

If we let a = the number of teachers teaching one class, b = the number of teachers teaching two classes, and c = the number of teachers teaching 3 classes, we can create the following equations:

a + b + c = 37
a + 2b + 3c = 64

Subtracting equation 1 from equation 2, we have:

(a + 2b + 3c = 64) - (a + b + c = 37)
b + 2c = 27
2c = 27 - b
c = (27 - b)/2

We see that c is the GREATEST when b = 1, and thus (27 - 1)/2 = 26/2 = 13.

We also see that c is the LEAST when b = 27, and thus (27 - 27)/2 = 0/2 = 0.

So, the range of values of n is 0 to 13.

Correct Answer: A

Approach Solution 3:
if n=13, then 13 teachers cover 39 classes.
Remaining classes are 64-39 = 25
Remaining teachers are 37-13 = 24, which is good as we have more classes than teachers.
if n=0, then we have 37 teachers covering 64 classes.

As these teachers can teach anywhere between 1-2, we know that we have enough teachers to cover all classes without any teacher teaching 3 classes.

Correct Answer: A

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