1997 AHSME Problems/Problem 18
Problem
A list of integers has mode 
and mean 
. The smallest number in the list is 
. The median 
of the list is a member of the list. If the list member were replaced by 
, the mean and median of the new list would be 
and , respectively. If were instead replaced by 
, the median of the new list would be 
. What is ?
Solution
Let there be 
integers on the list. The list of integers has mean , so the sum of the integers is 
.
Replacing with will increase the sum of the list from to 
.
The new mean of the list is , so the new sum of the list is also 
.
Thus, we get 
, leading to 
numbers on the list.
If there are 
numbers on the list with mode and smallest number , then the list is 
Since replacing with gives a new median of , and must be on the list of integers since is odd, 
, and the list is now 
The sum of the numbers on this list is 
, so we get:
, giving answer 
.
The original list is 
, with mean 
and median 
and mode .
The second list is 
, with mean 
and median 
.
The third list is 
with median 
.
See also
| 1997 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
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