2016 AMC 8

Complete problem set with solutions and individual problem pages

Problem 18 Hard

In an All-Area track meet, 216 sprinters enter a 100-meter dash competition. The track has 6 lanes, so only 6 sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?

  • A.

    36

  • B.

    42

  • C.

    43

  • D.

    60

  • E.

    72

Answer:C

Solution 1

From any n-th race, only \frac{1}{6} will continue on. Since we wish to find the total number of races, a column representing the races over time is ideal. Starting with the first race:

\frac{216}{6}=36

\frac{36}{6}=6

\frac{6}{6}=1

Adding all of the numbers in the second column yields \boxed{\textbf{(C)}\ 43}

 

Solution 2

Every race eliminates 5 players. The winner is decided when there is only 1 runner left. You can construct the equation: 216 - 5x = 1. Thus, 215 players have to be eliminated. Therefore, we need \frac{215}{5} games to decide the winner, or \boxed{\textbf{(C)}\ 43}

Related Problems
Problem 16 2016 AMC 8
Problem 17 2016 AMC 8
Problem 19 2016 AMC 8
Problem 20 2016 AMC 8
Think Academy Powered by ChatGPT