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A certain experimental mathematics program

This topic has 3 expert replies and 1 member reply
rsarashi Master | Next Rank: 500 Posts Default Avatar
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A certain experimental mathematics program

Post Sat Mar 25, 2017 9:19 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

OAA

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Post Thu Mar 30, 2017 3:29 pm
rsarashi 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
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.

Since n = the number of teachers teaching 3 classes, and if we let a = the number of teachers teaching 1 class and b = the number of teachers teaching 2 classes, we can create the following equations:

a + b + n = 37

a + 2b + 3n = 64

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

Since n is an integer, we see that n is the GREATEST when b = 1, and thus n = (27 - 1)/2 = 26/2 = 13 and n is the LEAST when b = 27. Thus, n = (27 - 27)/2 = 0/2 = 0.

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

Answer: A

_________________
Jeffrey Miller Head of GMAT Instruction

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Post Sat Mar 25, 2017 11:16 am
rsarashi 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

OAA
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

_________________
Brent Hanneson – Founder of GMATPrepNow.com
Use our video course along with Beat The GMAT's free 60-Day Study Guide

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Post Thu Mar 30, 2017 3:29 pm
rsarashi 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
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.

Since n = the number of teachers teaching 3 classes, and if we let a = the number of teachers teaching 1 class and b = the number of teachers teaching 2 classes, we can create the following equations:

a + b + n = 37

a + 2b + 3n = 64

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

Since n is an integer, we see that n is the GREATEST when b = 1, and thus n = (27 - 1)/2 = 26/2 = 13 and n is the LEAST when b = 27. Thus, n = (27 - 27)/2 = 0/2 = 0.

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

Answer: A

_________________
Jeffrey Miller Head of GMAT Instruction

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Post Sat Mar 25, 2017 11:16 am
rsarashi 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

OAA
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

_________________
Brent Hanneson – Founder of GMATPrepNow.com
Use our video course along with Beat The GMAT's free 60-Day Study Guide

Check out the online reviews of our course
Come see all of our free resources

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