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Math Help - Maximized Product with given sum

  1. #1
    Member Rimas's Avatar
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    Maximized Product with given sum

    Amoung all collevtions,S, of positive itegers whose sum is 28 what is the largest product that the intergers in s can form

    My teacher gives problems that she never teaches in school
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  2. #2
    excelmaths.com
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    Maximum value of product

    Collection S={ x,y are integers such that x,y>0 and x+y=28}

    Let P be the product of x and y => P=xy
    Now we have y=28-x => P=x.(28-x)
    =28x-(x^2)

    To find the largest value of the product P, differentiate P with respect to x and equate to zero.
    We get 28-2x=0
    => 28=2x
    => x=14
    thus y=28-x=14
    So the largest product =x.y=196
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by excelmaths.com View Post
    Collection S={ x,y are integers such that x,y>0 and x+y=28}

    Let P be the product of x and y => P=xy
    Now we have y=28-x => P=x.(28-x)
    =28x-(x^2)

    To find the largest value of the product P, differentiate P with respect to x and equate to zero.
    We get 28-2x=0
    => 28=2x
    => x=14
    thus y=28-x=14
    So the largest product =x.y=196
    The question does not restrict the size of S to two elements (it may have
    intended to but it does not).

    RonL
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  4. #4
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    Hello, Rimas!

    Among all collections, S, of positive integers whose sum is 28
    what is the largest product that the integers in S can form?

    The Captain is correct: the number of integers is not specified
    and the problem appears to be deliberately worded that way.

    It can be shown that, given a set of numbers with a fixed sum,
    . . the maximum product is achieved when the numbers are equal.

    Let n = number of integers. .Then each number is: 28/n
    . . Their product is: .P .= .(28/n)^n

    Since 28/n is an integer, n must be a divisor of 28.
    There are only six of them . . .

    n = 1: . P .= .28^1 .= . . .1

    n = 2: . P .= .14^2 .= . .196

    n = 4: . P .= . 7^4 . = . 2,401

    n = 7: . P .= . 4^7 . = .16,384 .

    n = 14: .P .= .2^14 .= .16,384 .

    n = 28: .P .= .1^28 .= . . .1


    Therefore, the largest possible product is 16,384.

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