Problem Solving

Joy Shaha wrote:Q. 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

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

Joy Shaha wrote:Q. 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

Sometimes the options can really help us a lot. Let us see how.

We see that there are 2*32 = 64 classes and 37 teachers who can take one, two or three classes.

Let us first find out what is the least value of n (the number of teachers who taught three classes)

Intuitively, we see that if each of the 37 teachers takes two classes, they together can take 2*37 = 74 classes > 64 required classes. So there is no need for even a single teacher to take three classes. Thus, the least value of n = 0. The correct option must be A or B.

Let's test option B since 14 is the maximum value between option A and option B.

Maximum value of n = 14.

=> Number of classes by n=14 teachers = 3*14 = 42.

=> Number of classes left = 64 - 42 = 22 and the number of teachers left = 37-14=23

Since 23 > 22, if 22 teachers take one class each, one teacher would be left taking no class; however, it is not possible since each teacher takes at least one class.

So the correct answer must be A.

Let's test option A for the sake of understanding.

Maximum value of n = 13.

=> Number of classes by n=13 teachers = 3*13 = 39.

=> Number of classes left = 64 - 39 = 25 and the number of teachers left = 37-13=24

So, we have to accommodate 25 classes by 24 teachers. Since 25 > 24, one teacher must have taken two classes.

Answer: A

Number of teachers who taught one class = 23
Number of teachers who taught two classes = 1
Number of teachers who taught three classes = 13

Hope this helps!

-Jay
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Joy Shaha wrote:Q. 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

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 n = the number of teachers teaching 3 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