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Thread: Possible Number of Students

  1. #1
    Super Member harpazo's Avatar
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    Possible Number of Students

    In a certain class consisting of 36 students, some boys anf some girls, exactly 1/3 of the boys and exactly 1/4 of the girls walk to school.

    This is what pops to mind:

    (1/3)(36) = 12 boys walk to school.

    (1/4)(36) = 9 girls walk to school.

    What is the greatest possible number of students in this class who walk to class?

    I decided to adf 9 and 12 to 21 students walk to school.
    Of course, my answer is wrong.
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  2. #2
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    Re: Possible Number of Students

    Quote Originally Posted by harpazo View Post
    In a certain class consisting of 36 students, some boys and some girls, exactly 1/3 of the boys and exactly 1/4 of the girls walk to school. This is what pops to mind:
    (1/3)(36) = 12 boys walk to school.
    (1/4)(36) = 9 girls walk to school.
    What is the greatest possible number of students in this class who walk to class?
    I decided to add 9 and 12 to 21 students walk to school.
    Of course, my answer is wrong.
    It is not that one fourth of the students are girls but one fourth of the girls walk to school.
    If there are $b$ boys in the class and there are $g$ girls in the class then $\bf{b+g=36}$.
    Thus $\dfrac{1}{3}b+\dfrac{1}{4}g$ is the number of students who walk.
    Now we cannot have a fraction of a person so $b$ must be divisible by $3$ and $g$ must be divisible by $4$.
    See what you can do with that.
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  3. #3
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    Re: Possible Number of Students

    Quote Originally Posted by harpazo View Post
    In a certain class consisting of 36 students, some boys anf some girls, exactly 1/3 of the boys and exactly 1/4 of the girls walk to school.

    This is what pops to mind:

    (1/3)(36) = 12 boys walk to school.
    That's assuming all 36 students are boys.

    (1/4)(36) = 9 girls walk to school.
    That's assuming all 36 students are girls.

    What is the greatest possible number of students in this class who walk to class?

    I decided to adf 9 and 12 to 21 students walk to school.
    Of course, my answer is wrong.
    Of course. We can't have "all 36 students are boys" and "all 36 students are girls"!

    Since the problem talks about " exactly 1/3 of the boys and exactly 1/4 of the girls" the number of boys must be a multiple of 3 and the number of girls must be a multiple of 4. Look at some possibilities:
    1) 0 girls: all 36 (a multiple or 3) students are boys. Then 36/3= 12 students walk to school.
    2) 4 girls: that would leave 36- 4= 32 boys which is not a multiple of 3.
    3) 8 girls: that would leave 36- 8= 28 boys which is not a multiple of 3.
    4) 12 girls: that would leave 36- 12= 24 boys which is a multiple of 3. (24/3)+ (12/4)= 8+ 3= 11 students walk to school.
    5) 16 girls: that would leave 36- 16= 20 boys which is not a multiple of 3.
    6) 20 girls: that would leave 36- 20= 16 boys which is not a multiple of 3.
    7) 24 girls: that would leave 36- 24= 12 boys which is not a multiple of 3. (12/3)+ (24/4)= 4+ 6= 10 students walk to school.
    8) 28 girls: that would leave 36- 28= 8 boys which is not a multiple of 3.
    9) 32 girls: that would leave 36- 32= 4 boys which is not a multiple of 3.
    10) 36 giris, 0 boys. 36/4= 9 walk to school.

    Now, can we interpret "a certain class consisting of 36 students, some boys and some girls" to mean that there are not 0 girls?
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  4. #4
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    Re: Possible Number of Students

    Girls must be multiple of 4, boys multiple of 3.
    List the possibilities:
    g b
    4 32
    8 28
    12 24 *
    16 20
    20 16
    24 12 *
    28 8
    32 4

    *: which one is greatest?

    Addendum:
    b>0 and g>0, else Mark and Deb will send me to the corner
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  5. #5
    Super Member harpazo's Avatar
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    Re: Possible Number of Students

    Quote Originally Posted by DenisB View Post
    Girls must be multiple of 4, boys multiple of 3.
    List the possibilities:
    g b
    4 32
    8 28
    12 24 *
    16 20
    20 16
    24 12 *
    28 8
    32 4

    *: which one is greatest?

    Addendum:
    b>0 and g>0, else Mark and Deb will send me to the corner
    Note: 12 24 is the greatest
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: Possible Number of Students

    The number of students that walk to school is:

    $\displaystyle W=\frac{1}{3}B+\frac{1}{4}G=\frac{4B+3G}{12}$

    Where:

    $\displaystyle B+G=36$

    Hence:

    $\displaystyle W=\frac{4B+3(36-B)}{12}=\frac{B+108}{12}=\frac{B}{12}+9$

    Since, as Denis points out there cannot be all 36 boys, we then take the number of boys to be 24 to get a maximum of 11 students that walk.
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  7. #7
    Super Member harpazo's Avatar
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    Re: Possible Number of Students

    Great notes from everyone. Thanks.
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  8. #8
    Super Member harpazo's Avatar
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    Re: Possible Number of Students

    Quote Originally Posted by MarkFL View Post
    The number of students that walk to school is:

    $\displaystyle W=\frac{1}{3}B+\frac{1}{4}G=\frac{4B+3G}{12}$

    Where:

    $\displaystyle B+G=36$

    Hence:

    $\displaystyle W=\frac{4B+3(36-B)}{12}=\frac{B+108}{12}=\frac{B}{12}+9$

    Since, as Denis points out there cannot be all 36 boys, we then take the number of boys to be 24 to get a maximum of 11 students that walk.
    I just love the way you create an equation from written information. Wish I had this skill under control.
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