2015 AMC 12B Problems/Problem 5

Problem

The Tigers beat the Sharks 2 out of the 3 times they played. They then played $N$ more times, and the Sharks ended up winning at least 95% of all the games played. What is the minimum possible value for $N$ ?

$\textbf{(A)}\; 35 \qquad \textbf{(B)}\; 37 \qquad \textbf{(C)}\; 39 \qquad \textbf{(D)}\; 41 \qquad \textbf{(E)}\; 43$

Solution

The ratio of the Shark's victories to games played is $\frac{1}{3}$ . For $N$ to be at its smallest, the Sharks must win all the subsequent games because $\frac{1}{3} < \frac{95}{100}$ . Then we can write the equation

$\frac{1+N}{3+N}=\frac{95}{100}=\frac{19}{20}$

Cross-multiplying yields $20(1+N)=19(3+N)$ , and we find that $N=\fbox{\textbf{(B)}\; 37}$ .

2015 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2015 AMC 12B Problems/Problem 5

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