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22 January, 19:44

When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list

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  1. 22 January, 20:37
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    The number of integers in the original list = 4.

    Step-by-step explanation:

    If the mean is m, x = the original sum of the list and n = the number of integers in the list, we have the system:

    m = x/n, m + 2 = (x + 15) / (n + 1) and m + 1 = (x + 16) / (n + 2).

    From the first equation x = mn

    so substituting for x in the other 2 equations:

    m + 2 = (mn + 15) / (n + 1) ... (1)

    m + 1 = (mn + 16) / (n + 2) ... (2)

    From equation (1):

    (m + 2) (n + 1) = mn + 15

    mn + m + 2n + 2 = mn + 15

    m + 2n + 2 = 15

    m + 2n = 13

    m = 13 - 2n.

    Now substitute for m in equation (2):

    13 - 2n + 1 = (n (13 - 2n) + 16) / (n + 2)

    14 - 2n = (13n - 2n^2 + 16) / (n + 2) Multiplying through by n + 2:

    (14 - 2n) (n + 2) = 13n - 2n^2 + 16

    14n + 28 - 2n^2 - 4n = 13n - 2n^2 + 16

    10n + 28 = 13n + 16

    12 = 3n

    n = 4.
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