2002 AMC 10B Problems/Problem 25
Contents
[hide]Problem
When is appended to a list of integers, the mean is increased by
. When
is appended to the enlarged list, the mean of the enlarged list is decreased by
. How many integers were in the original list?
Solution
Let be the sum of the integers and
be the number of elements in the list. Then we get the equations
and
. With a lot of algebra, the solution is found to be
.
Solution 2
We let be the original number of elements in the set and we let
be the original average of the terms of the original list. Then we have
is the sum of all the elements of the list. So we have two equations:
and
Simplifying both equations and we get,
Solving for
and
, we get
and
.
See also
2002 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.