Difference between revisions of "1997 AJHSME Problems/Problem 14"
Talkinaway (talk | contribs) (Created page with "==Problem== There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smal...") |
|||
| Line 28: | Line 28: | ||
* [[AJHSME Problems and Solutions]] | * [[AJHSME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
| + | {{MAA Notice}} | ||
Revision as of 23:26, 4 July 2013
Problem
There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?
Solution
Call the set 
, with 
.
Since the median, or middle number, is 
, we have 
.
Since the mode, or most common number, is 
, we have 
.
Thus, the set is now 
. Additionally, since there is only one mode, 
and 
are both less than and are distinct from each other.
Since the mean of the set is , and there are five numbers in the set, the sum of the numbers in the set is 
.
Thus, we have 
, which leads to 
Since and are distinct positive integers, they must equal 
and 
.
The smallest number is , and the largest number is , giving a difference of 
, leading to answer 
See also
| 1997 AJHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
