2021 AMC 10 A Fall

Complete problem set with solutions and individual problem pages

Problem 10 Easy

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 ?(2021 AMC Fall 10A, Question #10)

  • A.

    -18.5

  • B.

    -13.5

  • C.

    0

  • D.

    13.5

  • E.

    18.5

Answer:B

The formula for expected values is \text { Expected Value }=\sum(\text { Outcome } \cdot \text { Probability }) . We have \begin{aligned} t &=50 \cdot \frac{1}{5}+20 \cdot \frac{1}{5}+20 \cdot \frac{1}{5}+5 \cdot \frac{1}{5}+5 \cdot \frac{1}{5} \\ &=(50+20+20+5+5) \cdot \frac{1}{5} \\ &=100 \cdot \frac{1}{5} \\ &=20, \\ 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 \\ &=33.5 \end{aligned} Therefore, the answer is t-s=(\mathbf{B})-13.5.

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