Difference between revisions of "2021 Fall AMC 12A Problems/Problem 7"
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==Problem== | ==Problem== | ||
A school has <math>100</math> students and <math>5</math> teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are <math>50, 20, 20, 5, </math> and <math>5</math>. Let <math>t</math> be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let <math>s</math> be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is <math>t-s</math>? | A school has <math>100</math> students and <math>5</math> teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are <math>50, 20, 20, 5, </math> and <math>5</math>. Let <math>t</math> be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let <math>s</math> be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is <math>t-s</math>? | ||
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-Note that this problem only works under the assumption that there are no absences of students or teachers. | -Note that this problem only works under the assumption that there are no absences of students or teachers. | ||
Revision as of 11:53, 12 July 2024
- The following problem is from both the 2021 Fall AMC 10A #10 and 2021 Fall AMC 12A #7, so both problems redirect to this page.
Contents
Problem
A school has 
students and 
teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 
and . Let 
be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let 
be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is 
?
-Note that this problem only works under the assumption that there are no absences of students or teachers.
Solution
The formula for expected values is ![\[\text{Expected Value}=\sum(\text{Outcome}\cdot\text{Probability}).\]](http://latex.artofproblemsolving.com/d/5/0/d508273b090899391634032a9b333fb87f84fe09.png)
We have
Therefore, the answer is 
~MRENTHUSIASM
Video Solution (Simple and Quick)
~Education, the Study of Everything
Video Solution by TheBeautyofMath
for AMC 10: https://youtu.be/ycRZHCOKTVk?t=789
for AMC 12: https://youtu.be/wlDlByKI7A8?t=157
~IceMatrix
Video Solution by WhyMath
~savannahsolver
See Also
| 2021 Fall AMC 12A (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 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
| 2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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. 
