Difference between revisions of "2021 Fall AMC 10B Problems/Problem 9"

Revision as of 18:23, 24 October 2022

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 1

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 $2b = \frac{7}{27} = \boxed{\textbf{(C) }\frac{7}{27}}$

~KingRavi

Solution 2

We denote by $p$ the fraction of red knights who are magical.

Hence, $\frac{1}{6} = \frac{2}{7} p + \left( 1 - \frac{2}{7} \right) \frac{p}{2} .$

By solving this equation, we get $p = \frac{7}{27}$ .

Therefore, the answer is $\boxed{\textbf{(C) }\frac{7}{27}}$ .

~Steven Chen (www.professorchenedu.com)

Solution 3

Suppose there are $42$ knights. Then there are $\frac{2}{7}\cdot 42=12$ red knights and $42-12=30$ blue knights.

There are $\frac{1}{6}\cdot 42=7$ magical knights. Let $x$ be the number of red magical knights.

Then we have the equation $\frac{x}{12}=2\cdot \frac{7-x}{30}=\frac{7-x}{15}$ . Solving, we get $x=84/27$ .

Plugging $x$ back gives us the desired fraction of red magical knights, which is $\frac{\frac{84}{27}}{12}=\frac{84}{27}\cdot \frac{1}{12}=\boxed{\textbf{(C) }\frac{7}{27}}$ .

~DairyQueenXD

Video Solution by Interstigation

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

Video Solution

https://youtu.be/LJ9mxTNSRUU

~Education, the Study of Everything

Video Solution by WhyMath

https://youtu.be/Ia0Jbe1h_qY

~savannahsolver

@@ Line 34: / Line 34: @@
 == Solution 3 ==
-Suppose there are 42 knights. Then there are <math>\frac{2}{7}\cdot 42=12</math> red knights and <math>42-12=30</math> blue knights.
+Suppose there are <math>42</math> knights. Then there are <math>\frac{2}{7}\cdot 42=12</math> red knights and <math>42-12=30</math> blue knights.
 There are <math>\frac{1}{6}\cdot 42=7</math> magical knights. Let <math>x</math> be the number of red magical knights.

2021 Fall AMC 10B (Problems • Answer Key • Resources)
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