Difference between revisions of "2021 Fall AMC 12A Problems/Problem 7"
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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>? | ||
− | <math>\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ | + | <math>\textbf{(A)}\ {-}18.5 \qquad\textbf{(B)}\ {-}13.5 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 13.5 \qquad\textbf{(E)}\ 18.5</math> |
− | |||
==Solution== | ==Solution== | ||
Line 11: | Line 10: | ||
We have | We have | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | t &= \frac15\ | + | t &= 50\cdot\frac15 + 20\cdot\frac15 + 20\cdot\frac15 + 5\cdot\frac15 + 5\cdot\frac15 \\ |
− | &= | + | &= (50+20+20+5+5)\cdot\frac15 \\ |
− | &= \frac15 | + | &= 100\cdot\frac15 \\ |
&= 20, \\ | &= 20, \\ | ||
− | s &= \frac{50}{100}\ | + | s &= 50\cdot\frac{50}{100} + 20\cdot\frac{20}{100} + 20\cdot\frac{20}{100} + 5\cdot\frac{5}{100} + 5\cdot\frac{5}{100} \\ |
&= 25 + 4 + 4 + 0.25 + 0.25 \\ | &= 25 + 4 + 4 + 0.25 + 0.25 \\ | ||
&= 33.5. | &= 33.5. | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | Therefore, the answer is <math>t-s=\boxed{\textbf{(B)}\ | + | Therefore, the answer is <math>t-s=\boxed{\textbf{(B)}\ {-}13.5}.</math> |
~MRENTHUSIASM | ~MRENTHUSIASM | ||
+ | |||
+ | ==Video Solution (Simple and Quick)== | ||
+ | https://youtu.be/nZBIoUS0mdk | ||
+ | |||
+ | ~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== | ||
+ | https://youtu.be/f7vhOCnvl0k | ||
+ | |||
+ | ~savannahsolver | ||
==See Also== | ==See Also== |
Latest revision as of 03:02, 13 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
?
Solution
The formula for expected values is
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.