2021 Fall AMC 10B Problem 9

Problem 9 of 25EasierAlgebra

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?

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Solution

Let

x

be the fraction of blue knights who are magical. Then the fraction of red knights who are magical is

2x.

The total magical fraction is a weighted average over the red and blue groups:

\frac27(2x)+\frac57(x)=\frac16.

Thus

\frac97x=\frac16,

x=\frac7{54}.

The red magical fraction is

2x=\frac7{27}.

Thus, the answer is C .