Difference between revisions of "2013 AIME II Problems/Problem 7"
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Revision as of 13:54, 4 July 2013
Problem 7
A group of clerks is assigned the task of sorting 
files. Each clerk sorts at a constant rate of 
files per hour. At the end of the first hour, some of the clerks are reassigned to another task; at the end of the second hour, the same number of the remaining clerks are also reassigned to another task, and a similar assignment occurs at the end of the third hour. The group finishes the sorting in 
hours and 
minutes. Find the number of files sorted during the first one and a half hours of sorting.
Solution
There are 
clerks at the beginning, and 
clerks are reassigned to another task at the end of each hour. So, 
, and simplify that we get 
.
Now the problem is to find a reasonable integer solution. Now we know 
, so 
divides 
, as long as is a integer, must divide 
. Now, we suppose that 
, similarly we get 
, and so in order to get a minimum integer solution for , it is obvious that 
works. So we get 
and 
. One and a half hour's work should be 
, so the answer is 
.
See Also
| 2013 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
