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Math Help - Harmonic mean?

  1. #1
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    Harmonic mean?

    Find two positive numbers that differ by 12 and have harmonic mean 5...

    I started out with 1/2(1/x+12 + 1/x-12) = 1/5 but then it got all jumbled
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  2. #2
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    Quote Originally Posted by Dunit0001 View Post
    Find two positive numbers that differ by 12 and have harmonic mean 5...

    I started out with 1/2(1/x+12 + 1/x-12) = 1/5 but then it got all jumbled
    The two positive integers:
    Let x = one of them
    And y = the other, and greater than x.
    So,
    y -x = 12
    y = x +12 ---------**

    Their Harmonic Mean,
    H = 2 / [1/x +1/(x+12)] = 5

    2 / [1/x +1/(x+12)] = 5
    2/5 = 1/x +1/(x+12)
    Clear the fractions, multiply both sides by 5(x)(x+12),
    2(x)(x+12) = 1(5)(x+12) +1(5)(x)
    2x^2+24x = 5x +60 +5x
    2x^2 +24x -10x -60 = 0
    2x*2 +14x -60 = 0
    x^2 +7x -30 = 0
    (x +10)(x -3) = 0
    x = -10 or 3

    Positive integer, so x = 3
    y = 3 +12 = 15

    Check,
    2 / [1/3 +1/15] =? 5
    2 / [(15 +3)/(3*15)] =? 5
    2 / [18 / 45] =? 5
    2(45/18) =? 5
    90/18 =/ 5
    5 =? 5
    Yes, so, OK.

    Therefore, the two positive integers are 3 and 15. --------answer.
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