2021 AMC Fall 10A Problem 10

1. Find $t$. Each teacher reports the size of their own class, and each teacher is equally likely. So $t$ is the ordinary average of the five numbers:
$$t = \frac{50 + 20 + 20 + 5 + 5}{5} = \frac{100}{5} = 20.$$
2. Find $s$: what does each student say? Every student reports the size of their own class. List what all $100$ students say:
- The $50$ students in the big class each say $50$. Together: $50 \times 50 = 2500$.
- The $20 + 20 = 40$ students in the two middle classes each say $20$. Together: $40 \times 20 = 800$.
- The $5 + 5 = 10$ students in the two small classes each say $5$. Together: $10 \times 5 = 50$.
3. Average those $100$ answers.
$$s = \frac{2500 + 800 + 50}{100} = \frac{3350}{100} = 33.5.$$
4. Subtract. $t - s = 20 - 33.5 = -13.5$.

The answer is $\boxed{\textbf{(B)}\ {-}13.5}$.

When you average over teachers, each class counts once. When you average over students, each class counts once for every student in it, so the big class gets a much louder voice. That is why $s$ is bigger than $t$, and you can tell before computing that $t - s$ must be negative, leaving only (A) or (B). This is the reason students often feel their classes are bigger than the school's "average class size".

For $t$, each of the $5$ teachers is equally likely, so $t$ is the ordinary mean of the five class sizes:
$$t = \frac{50 + 20 + 20 + 5 + 5}{5} = \frac{100}{5} = 20.$$

For $s$, each of the $100$ students is equally likely, and a student in a class of size $n$ reports $n$. There are $50$ students who report $50$, $40$ who report $20$, and $10$ who report $5$:
$$s = \frac{50 \cdot 50 + 20 \cdot 20 + 20 \cdot 20 + 5 \cdot 5 + 5 \cdot 5}{100} = \frac{2500 + 800 + 50}{100} = 33.5.$$

Therefore $t - s = 20 - 33.5 = -13.5$.

The answer is $\boxed{\textbf{(B)}\ {-}13.5}$.