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

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.