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Math Help - Sets / Counting Problem

  1. #1
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    Sets / Counting Problem

    Here is the problem.

    70 students are enrolled in Math, English, or German. 40 students are in Math, 35 are in English, 30 are in German. 15 students are enrolled in all 3 of the courses. How many of the students are enrolled in exactly two of the courses: Math, English, and German?

    The book has very bad explanation and possibly wrong answer for this problem. Using my own logic I get a different answer than the book.

    Please let me know what you guys get as answer for this problem.

    thank you.


    P.S. Anyone preparing for GMAT, please PM me if you're good at math. We might be able to help eachother out! Thanks.
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  2. #2
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    Draw a picture: Draw 3 circles, overlapping. Each circle represents the students that are taking that particular course. Label each region as "Math", "German", and "English. In that region where all three circles overlap write the number 15 to represent the 15 students that are taking all three courses. In the region where the "English" and "Math" circles overlap write "x" representing the (unknown) students taking those two courses. In the region where "Math" and "German" overlap, write "y" representing the (unknown) students taking those two courses. In the region where the "English" and "German" circles overlap, write "z" representing the (unknown) students taking those two courses. Since there are a total of 40 students taking math there must be 40- (x+ y+ 15)= 25- x- y in the final open region of the "Math" circle- representing those student who are taking only math. Since there are 35 students taking English, there must be 35- (x+z+ 15)= 20- x- z in the final open region of the "English" circle- representing those students who are taking only English. Since there are 30 students taking German, the must be 30- (y+ z+ 15)= 15- y- z in the final open region of the "German" circle- representing those students who are taking only German.

    The point of all that is that each student is counted in that only once. We can get the total number of students by adding those numbers:
    (25-x-y)+ (20-x-z)+ (15-y-z)+ (x+ y+ z)+ 15= 75- (x+ y+ z) and we know there are a total of 70 students!


    where , "y", and "z" representing the (unknown) students taking exactly two of the courses. In the final 3 regions write
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  3. #3
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    Thank you for your response. Is there a shorter way of doing this problem? The equation you have created has 2 unknowns, and we need a numerical answer.

    Here is what I did. I added all the students taking Math, English, and German and subtracted 15 because that is number taking all 3 courses:

    40 + 35 + 30 - 15 = 90

    Therefore, 90 is the number of students taking 1 course or 2 courses.

    The question is a little confusing but I think "70" refers to the number of students taking only 1 course; either math, english, or german.

    Therefore 90 - 70 = 20 (which is my answer).

    Using your equation, tell me what answer you get as I am really bad with these types of equations.

    Thanks!
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  4. #4
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    ??? The equation I gave you was 75- (x+y+z)= 70 and you can easily get x+y+z= 5 from that. The problem asked "How many of the students are enrolled in exactly two of the courses"!
    x is the number of students enrolled in English and Math and only those two courses.
    y is the number of students enrolled in Math and German and only those two courses.
    z is the number of students enrolled in German and English and only those two courses.

    So how many students are enrolled in exactly two courses?
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  5. #5
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    you are right... sorry I got confused...
    thank you for your help.

    too used to solving for one variable that I can't think about solving for sum of 3.
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