2021 Fall AMC 10B Problems/Problem 9

Problem

The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical?

$\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{3}{13}\qquad\textbf{(C) }\frac{7}{27}\qquad\textbf{(D) }\frac{2}{7}\qquad\textbf{(E) }\frac{1}{3}$

Solution

Let $k$ be the number of knights: then the number of red knights is $\frac{2}{7}k$ and the number of blue knights is $\frac{5}{7}k$ .

Let $b$ be the fraction of blue knights that are magical - then $2b$ is the fraction of red knights that are magical. Thus we can write the equation $b \cdot \frac{5}{7}k + 2b \cdot \frac{2}{7}k = \frac{k}{6}\implies \frac{5}{7}b + \frac{4}{7}b = \frac{1}{6}$ $\implies \frac{9}{7}b = \frac{1}{6} \implies b=\frac{7}{54}$

We want to find the fraction of red knights that are magical, which is $2b = \frac{7}{27} = \boxed{C}$

~KingRavi

Video Solution by Interstigation

https://youtu.be/p9_RH4s-kBA?t=1274

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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2021 Fall AMC 10B Problems/Problem 9

Contents

Problem

Solution

Video Solution by Interstigation

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