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Intuition behind this algebraic question

  1. Apr 5, 2009 #1
    Hi PF members!

    I'd like some guidance with the intuition behind the following problem. I have provided my intuition as far as I can below but if anyone can help clarify it I'd be very thankful.

    Question: Tickets to a concert cost $9.00 for adults and $6.50 for students. A total of 950 people paid $7550 to attend. How many students attended the concert?

    I. The Algebra
    Code (Text):

    The answer basically involves solving two equations with two unknowns:

    s = students, a = adults

    i) a + s = 950 and 9a + 6.5s = 7550

    ii) a = 950 - s

    iii) sub (ii) into 9a in (i)

    iv) 8550 - 7550 = 2.5s

    v) s = 400 (?)

     
    II. My intuition (logic) behind the algebra


    i) total # of people who came (students and adults) equals 950. Revenue from adults is ($9 per adult)*(total number of adults = a). Revenue from students is ($6.50 per student)*(total number of students = s)

    ii) if in total 950 people came and assuming we know how many students came then those that are not students must be adults. In other words, a = 950 - s

    iii) 9(950-s) + 6.5(s) = 8550 - 9s + 6.5s = 7550

    That is, assume that we charge all 950 people $9. This revenue comes to $8550. Then we take out the students to whom we charged $9, whom are -9s. Then we will be left on one hand with the adults to whom we charged $9 (8550 - 9s). Plus we add the students to whom we charge $6.5 (6.5s). Sum will equal the actual revenue of $7550.


    iv) 8550-7550 = 1000 = 2.5s

    The revenue that comes from charging students AND adults $9 minus the actual revenue of $7550 from charging ONLY adults $9 gives $1000, which equals the extra $2.5 that all students WOULD'VE paid HAD we charged them $9. How many such students are there? 1000/2.5 = 400.


    So, I need your help with iv in part II. How do I interpret equation iv? And that, too, logically?
     
  2. jcsd
  3. Apr 5, 2009 #2
    It looks like you've got it, what do you need help with?
     
  4. Apr 5, 2009 #3
    How do you interpret in your words 2.5s = 1000?

    That is, within the context of this problem, how do you interpret $2.5 per student times some number of students equals $1000?
     
    Last edited: Apr 5, 2009
  5. Apr 6, 2009 #4
    lifelearner,

    Am I correct in assuming that you did not find the algebra, but are rather looking at the logic behing the algebra? If so:

    Part iv has skipped a couple of steps to get where it is. Basic algebra. Looking at step iii we get:

    a = 950 - s
    This can be altered to get,
    9a = 8550 - 9s
    Then substituting into equation i,
    8550 - 9s + 6.5s = 7550
    Simplify,
    8550 - 2.5s = 7550
    8550 = 7550 + 2.5s
    8550 - 7550 = 2.5s
    2.5s = 1000
    s = 400

    To find a,
    a = 950 - s
    a = 950 - 400
    a = 550

    And checking,
    9a + 6.5s = 7550
    9*550 + 6.5*400 = 7550
    4950 + 2600 = 7550


    So there you go, the 2.5s merely arises through algebra.

    Now, if you did find this already, and you worked out the algebra... I don't know what the problem is :smile:
     
  6. Apr 6, 2009 #5
    I understand completely the algebra. However, the physical meaning of 2.5s is what I don't understand. Somehow I feel Venn diagrams might be useful; i.e: 8550 is revenue you get if you charge all persons $9. When you subtract 9s then you effectively "filter" out the students whom you charge $9, which will leave you with revenue from adults, each of whom paid $9.

    I understand 2.5s arises through algebra but can we put some meaning behind it?
     
  7. Apr 6, 2009 #6
    I suppose the meaning might be 2.5 times the number of students is 1000?
     
  8. Apr 6, 2009 #7
    What is the 1000 mean in physical meaning? And of what significance is 2.5 times # of students?
     
  9. Apr 6, 2009 #8

    arildno

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    Why should it mean anything at all, other than being provably logically equivalent to a previous statement?

    You could multiply that equation with, say, 3.72 and you'd still have a logically equivalent equation, but it wouldn't have much "meaning" in the sense you are seeking.
     
  10. Apr 6, 2009 #9
    Should it not mean something as we are dealing with a practical question?
     
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