# 2016 AMC 8 Problems/Problem 18

18. 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?

$(A)\mbox{ }36\mbox{ }(B)\mbox{ }42\mbox{ }(C)\mbox{ }43\mbox{ }(D)\mbox{ }60\mbox{ }(E)\mbox{ }72\mbox{ }$

## Solution

From any nth 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 $43 \rightarrow \boxed{C}$

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