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Math Help - square problem

  1. #1
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    square problem

    The rows of chairs in section A of an auditorium are arranged in a square with as many chairs in each row as the total number of rows. Section B is also arranged in a similar manner. If section A has 23 more chairs than section B, how many chairs are in section A?

    It has to be 144 but why?
    Last edited by mr fantastic; March 19th 2010 at 06:13 PM. Reason: Restored deleted question
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  2. #2
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    Hi Sarah,

    The problem is to find a where a^2 - b^2 = 23. So you have 1 equation and two unknowns and this is a real problem! The only way I can see to solve it is to try some numbers. You suggested 144 i.e. a = 12^2.
    So b^2 = 144 - 23 = 121 and this is 11^2.

    I hope this helps
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  3. #3
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    Following on s_ingram's hint, a^2- b^2= (a- b)(a+ b)= 23.

    Since a and b are "numbers of chairs" they must be positive integers and the only way to factor 23 in positive integers is 1*23.

    Solve a- b= 1, a+ b= 23.
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    Following on s_ingram's hint, a^2- b^2= (a- b)(a+ b)= 23.

    Since a and b are "numbers of chairs" they must be positive integers and the only way to factor 23 in positive integers is 1*23.

    Solve a- b= 1, a+ b= 23.
    Thanks Ivy! I spotted it now. And thanks s_ingram too!
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  5. #5
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    Hello, Sarah!

    Knowing a fact about squares, I "eyeballed" the problem.


    Two consecutive squares always differ by an odd number.
    . . (n+1)^2 - n^2 \:=\:2n+1


    Since the difference is 23, we have: . 2n+1 \:=\: 23 \quad\Rightarrow\quad n = 11

    Therefore, the squares must be: . 11^2 and 12^2.

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