Question

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?

$\dfrac{2}{9}$

$\dfrac{3}{13}$

$\dfrac{7}{27}$

$\dfrac{2}{7}$

$\dfrac{1}{3}$

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 $\frac97x=\frac16$ , so $x=\frac7{54}$ . The red magical fraction is $2x=\frac7{27}$ .

Thus, the answer is C .

2021 AMC 10B Fall Problem 9

Solution:

Problem 9 in Other Years