Question

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 $t - s?$

$-18.5$

$-13.5$

$0$

$13.5$

$18.5$

Answer: B

Concepts:mean expected value

Difficulty rating: 1410

Solution:

The teacher average is $t = \dfrac{50 + 20 + 20 + 5 + 5}{5}$ $= \dfrac{100}{5} = 20.$

The student average weights each class size by how many students are in it: $\begin{aligned} s &= \small \frac{50^2 + 20^2 + 20^2 + 5^2 + 5^2}{100} \\ &= \small \frac{2500 + 400 + 400 + 25 + 25}{100} \\ &= 33.5. \end{aligned}$

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

Thus, the correct answer is B.

2021 AMC 12A Fall Problem 7

Answer: B

Difficulty rating: 1410

Solution:

Problem 7 in Other Years