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Math Help - which number is larger?

  1. #1
    Senior Member sfspitfire23's Avatar
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    which number is larger?

    Fellas, I have a question regarding quantitative comparisons.

    Say we have, in column A, (78)(243) and in column B, (77)(244).

    I'm asked to determine which is larger.

    Is it a good strategy to pick out the first two numbers in each variable? So, in column A we would take (78)(24) and B would be (77)(24). Thus A is larger because 78>77.

    What about A= (90,021)(100,210) and B=(90,210)(100,021)? Using my method, we have (900)(100) in A and (902)(100) in B. Thus, B is larger.

    Is it possible to do this? Is there some law?
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  2. #2
    A Plied Mathematician
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    What does the notation (78)(243) mean? I mean, I normally think usual arithmetic product of the two numbers. But in number theory, you might have a different context.
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  3. #3
    Senior Member sfspitfire23's Avatar
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    Sorry, it is the product
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  4. #4
    A Plied Mathematician
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    Ok, so I would say that, given two numbers whose sum is a constant, their product is maximized when the two numbers are closer together (global max is when they are equal). So I would say instantly that the product in Column A is greater. You can prove this using calculus: assume x+y=\text{constant}=c. The goal is to maximize the product xy=x(c-x). So, let

    f(x)=x(c-x). Then

    f'(x)=c-2x. Setting this equal to zero implies

    c-2x=0, or x=c/2=y. Done.
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  5. #5
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by sfspitfire23 View Post
    Fellas, I have a question regarding quantitative comparisons.

    Say we have, in column A, (78)(243) and in column B, (77)(244).

    I'm asked to determine which is larger.

    Is it a good strategy to pick out the first two numbers in each variable? So, in column A we would take (78)(24) and B would be (77)(24). Thus A is larger because 78>77.

    What about A= (90,021)(100,210) and B=(90,210)(100,021)? Using my method, we have (900)(100) in A and (902)(100) in B. Thus, B is larger.

    Is it possible to do this? Is there some law?
    (78)(243) = (77 + 1)(243) = 77 * 243 + 243

    (77)(244) = (77)(243 + 1) = 77 * 243 + 77
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  6. #6
    Senior Member sfspitfire23's Avatar
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    Very nice I get it, thanks!
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  7. #7
    MHF Contributor Also sprach Zarathustra's Avatar
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    Try to prove the following for a, b integers greater than 1.

    a*b>(a-1)*(b+1)
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  8. #8
    A Plied Mathematician
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    You're welcome. undefined and AsZ's methods are also both entirely valid. AsZ's method might be generalized by proving that

    ab > (a-m)(b+m).
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  9. #9
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Also sprach Zarathustra View Post
    Try to prove the following for a, b integers greater than 1.

    a*b>(a-1)*(b+1)
    You forgot to mention the restriction \displaystyle a \le b. (Edit: Possibly it was an intentional omission though.)
    Last edited by undefined; July 28th 2010 at 07:47 AM.
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