Anonymous
Anonymous asked in Science & MathematicsMathematics · 9 months ago

math homework?

A set of 10 positive numbers is combined with another set of 12 positive numbers. If the mean of the first set is 16, what is the third smallest possible mean of the second set so that the average of the combined set is an integer?

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  • Ian H
    Lv 7
    9 months ago

    If the mean of 10 positive numbers is 16 their total is 160. 

    If the mean of 12 positive numbers is m their total is 12m 

    The combined total of the 22 positive numbers is 12m + 160 

    The average of the combined 22 positive numbers, A = (12m + 160)/22 

    A = 7 + (12m + 6)/22 = 7 + (6m + 3)/11 

    Let the second term in that be another integer, i 

    m = (11i – 3)/6, so i = 3, gives us m = 5 as the lowest value of the mean m 

     

    But we needed the THIRD smallest possible mean of the second set. 

    To find that, we need to increase i by two steps of 6. 

    i = 3 + 12 = 15 which produces m = (165 – 3)/6 = 27

    The third smallest possible mean of the second set  was 27

     

  • 9 months ago

    The sum of the first set of 10 positive numbers must be 10*16 = 160.

    The average of the combined set would then be (160+12m) / 22, where m is the mean of the second set of 12 numbers.

    For the average of the combined set to be an integer, m must be 5, 16, 27, 38, 49...

    The third smallest possible mean of the second set of numbers would be 27.

  • The first set sums to 160.  Let m be the mean of the 2nd set

    (160 + 12m) / (10 + 12) = k

    k is an integer

    (160 + 12m) / 22 = k

    160 + 12m = 22k

    80 + 6m = 11k

    80 + 6m is a multiple of 11

    77 + 3 + 6m = 11k

    7 + 3 * (1 + 2m) / 11 = k

    (3/11) * (1 + 2m)

    m = 5 works

    m = 16 works

    m = 27 works

    (3/11) * (1 + 2 * (5 + 11j))

    (3/11) * (1 + 10 + 22j) =>

    (3/11) * (11 + 22j)

    (3/11) * 11 * (1 + 2j)

    3 * (1 + 2j)

    3 + 6j

    7 + 3 + 6j = k

    10 + 6j = k

    j is an integer

    j = 0 , 1 , 2 , 3 , 4 , 5 , ......

    10 + 6 * 0 , 10 + 6 * 1 , 10 + 6 * 2 , ....

    10 , 16 , 22 , 28 , 34 , 40 , 46 , 52 , 58 , 64 , 70 , ...

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