A certain experimental mathematics program was tried out in 2 classes in each of 32 elementary schools and involved 37

Gmat_mission wrote: ↑

Fri Jan 08, 2021 4:15 am

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

Answer: A

Source: Official Guide

Check the answer choices (ALWAYS check the answer choices before choosing a particular solution strategy)
I see that, for each answer choice, the second value (the greatest value of n) is different. So, let's test some of these values.

Let's start by testing answer choice B (0 and 14)
I'd like to start here, since we're asked to identify the greatest value of n, and answer choice B has the biggest possible value of n.
So, is it possible to have 14 teachers who teach 3 classes?
Well, (14)(3) = 42 classes
There are 64 classes altogether (2 classes in each of the 32 schools, means a total of 64 classes)
So, the number of classes that still require teachers = 64 - 42 = 22

How many teachers are remaining?
So far, 14 of the 37 teachers are accounted for (they're the ones who are teaching 3 classes each)
So, the number of teachers remaining = 37 - 14 = 23
Can these 23 remaining teachers cover the remaining 22 classes?
NO!
Each teacher must teach AT LEAST ONE class. So, there aren't enough classes needed for each teacher to teach at least one class.
So, we can ELIMINATE answer choice B.

IMPORTANT: We were VERY CLOSE with answer choice B. We were just one class short of meeting our goal. So, I am quite confident that the greatest possible values of n is 13 (answer choice A). Let's find out.

We'll test answer choice A (0 and 13)
Well, (13)(3) = 39 classes
There are 64 classes altogether
So, the number of classes that still require teachers = 64 - 39 = 25

So far, 13 of the 37 teachers are accounted for. So, the number of teachers remaining = 37 - 13 = 24
Can these 24 remaining teachers cover the remaining 25 classes?
YES!
23 of the teachers can teach 1 class each, and the other teacher can teach 2 classes.
Since the greatest possible value of n is 13, the correct answer is ...A


Cheers,
Brent

Gmat_mission wrote: ↑

Fri Jan 08, 2021 4:15 am

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

Answer: A

Source: Official Guide

Solution:

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 in this program.

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

a + b + n = 37. This is the equation for the total number of teachers.

a + 2b + 3n = 64. This is the equation for the number of classes taught by the teachers.

Subtracting equation 1 from equation 2, we have:

(a + 2b + 3n = 64) - (a + b + n = 37)

b + 2n = 27

2n = 27 - b

n = (27 - b)/2

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

We also see that n 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.

Answer: A