2021 AMC 10A Fall Problem 10

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

The teacher average weights each class once

The student average weights a class by its enrollment

Recall the expected-value formula

\mathbb E[X]=\sum_x x\,\Pr(X=x).

Therefore,

t = \dfrac{1}{5} (50 + 20 + 20 + 5 + 5)

= \dfrac{1}{5} \cdot 100 = 20

and

s = 50 \cdot \dfrac{50}{100} + 20 \cdot \dfrac{20}{100}

+ 20 \cdot \dfrac{20}{100} + 5 \cdot \dfrac{5}{100} + 5 \cdot \dfrac{5}{100}

\begin{aligned} &= 25 + 4 + 4 + 0.25 + 0.25 \\ &= 33.5 \end{aligned}

t - s = -13.5.

Thus, B is the correct answer.

Problem 10 in other years