Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2021 Fall AMC 12A Problems/Problem 7"

m (→‎Solution 1)
(→‎Solution 2: Removed repetitive solution. Prof. Chen agreed to this through PM ...)
Line 21: Line 21:
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM
−
 
−
== Solution 2 ==
 
−
First,
 
−
<cmath>
 
−
\begin{align*}
 
−
t & = \frac{50 + 20 + 20 + 5 + 5}{5} \\
 
−
& = 20 .
 
−
\end{align*}
 
−
</cmath>
 
−
 
−
 
−
Second, we compute <math>s</math>.
 
−
We have
 
−
<cmath>
 
−
\begin{align*}
 
−
s & = \frac{50}{100} \cdot 50 + \frac{20}{100} \cdot 20 + \frac{20}{100} \cdot 20
 
−
+ \frac{5}{100} \cdot 5 + \frac{5}{100} \cdot 5 \\
 
−
& = 33.5 .
 
−
\end{align*}
 
−
</cmath>
 
−
 
−
Therefore, <math>t - s = 20 - 33.5 = - 13.5</math>.
 
−
 
−
 
−
Therefore, the answer is <math>\boxed{\textbf{(B) }-13.5}</math>.
 
−
 
−
~Steven Chen (www.professorchenedu.com)
 
  
 
==See Also==
 
==See Also==

Revision as of 01:48, 26 November 2021

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.

Problem

A school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ 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 $t-s$?

$\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ {-}13.5 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 13.5 \qquad\textbf{(E)}\ 18.5$

Solution

The formula for expected values is \[\text{Expected Value}=\sum(\text{Outcome}\cdot\text{Probability}).\] We have \begin{align*} t &= \frac15\cdot50 + \frac15\cdot20 + \frac15\cdot20 + \frac15\cdot5 + \frac15\cdot5 \\ &= \frac15\cdot(50+20+20+5+5) \\ &= \frac15\cdot100 \\ &= 20, \\ s &= \frac{50}{100}\cdot50 + \frac{20}{100}\cdot20 + \frac{20}{100}\cdot20 + \frac{5}{100}\cdot5 + \frac{5}{100}\cdot5 \\ &= 25 + 4 + 4 + 0.25 + 0.25 \\ &= 33.5. \end{align*} Therefore, the answer is $t-s=\boxed{\textbf{(B)}\ {-}13.5}.$

~MRENTHUSIASM

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_Fall_AMC_12A_Problems/Problem_7&oldid=166509"