2002 AMC 10B Problem 25

Question

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?

4

5

6

7

8

Let the list have

n

integers with mean

m,

so the sum is

mn

Translate each appending into an equation for the new sum and mean

Let the original list have

n

integers with mean

m,

so its sum is

mn.

Appending

15

gives

\begin{aligned} &(m + 2)(n + 1) \\ &= mn + 15 \\ &\implies m + 2n = 13. \end{aligned}

Appending

1

to that enlarged list gives

\begin{aligned} &(m + 1)(n + 2) \\ &= mn + 16 \\ &\implies 2m + n = 14. \end{aligned}

Solving

m + 2n = 13

and

2m + n = 14

yields

m = 5

and

n = 4.

Thus, the correct answer is A.

Problem 25 in other years