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A certain experimental mathematics program

This topic has 3 expert replies and 1 member reply

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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
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    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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    Marty Murray Legendary Member
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    Post Sat Mar 25, 2017 9:43 am
    Number of Schools: 32

    Number of Classes Per School: 2

    Total Number of Classes: 32 x 2 = 64

    Number of Teachers: 37

    Each teacher teaches at least 1 class. So 37 classes are covered.

    There are 64 - 37 = 27 remaining.

    To maximize the number of teachers teaching 3 classes, assign as many as possible of the remaining 27 classes to teachers teaching 3 classes.

    First Assignment of Classes: 1 class each to 37 teachers

    Second Assignment of Classes: give 2 more classes to as many teachers as possible. Each teacher assigned 2 more classes will be teaching 3 classes.

    27/2 = 13, remainder 1.

    So we can assign 13 teachers 2 more classes each. So out of the 37 teachers the most who can have 3 classes is 13.

    So we have 13 teaching 3, 1 teaching 2, and 23 teaching 1.

    Maximum Teaching 3 Classes: 13

    To find the the minimum number that could be assigned 3 classes, use up as many as possible of the 27 left by assigning 1 more to each teacher, creating teachers who are teaching 2 classes.

    First Assignment of Classes: 1 class each to 37 teachers

    Second Assignment of Classes: 1 more class to each of 27 teachers

    Now all of the classes are assigned.

    So 27 teach 2 classes and the remaining 10 teach 1 class.

    0 teachers teach 3.

    Minimum Teaching 3 Classes: 0

    The correct answer is A.

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    Thanked by: rsarashi
    Post Sat Mar 25, 2017 10:36 am
    Hi rsarashi,

    The answer choices to this question are numbers, so we can use them 'against' the prompt and TEST THE ANSWERS.

    From the first sentence, we know that there are (2)(32) = 64 classes and 37 teachers. We're told that each teacher taught 1 to 3 of the classes and we're essentially asked for the 'range' of teachers who COULD have taught 3 classes.

    Let's start with the MINIMUM possibility - from the answer choices, it's 0, 1 or 2 teachers... so let's see if we can do it with 0...

    IF....
    0 teachers taught 3 classes...
    then 37 teachers taught 64 classes
    If 27 teachers taught 2 each = 54 classes
    and 10 teachers taught 1 each = 10 classes
    then all 64 classes could be taught without any teacher covering 3 classes.
    The minimum IS 0.
    Eliminate Answers C, D and E.

    Now, let's consider the MAXIMUM possibility - from the remaining answers, it's either 13 or 14 teachers... so let's see if we can do it with 14...

    IF....
    14 teachers taught 3 classes...
    then those 14 teachers taught (14)(3) = 42 classes...
    leaving 64 - 42 = 22 classes for the remaining 23 teachers....
    However, each of the 23 teachers would NOT get a class under these conditions...
    So the maximum is NOT 14.
    Eliminate Answer B.

    Final Answer: A

    GMAT assassins aren't born, they're made,
    Rich

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    Thanked by: rsarashi
    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

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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

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    Thanked by: rsarashi

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