Difference between revisions of "2021 Fall AMC 10B Problems/Problem 9"
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<math>\implies \frac{9}{7}b = \frac{1}{6} \implies b=\frac{7}{54}</math> | <math>\implies \frac{9}{7}b = \frac{1}{6} \implies b=\frac{7}{54}</math> | ||
| − | We want to find the fraction of red knights that are magical, which is <math>2b = \frac{7}{27} = \boxed{C}< | + | We want to find the fraction of red knights that are magical, which is <math>2b = \frac{7}{27} = </math>\boxed{\textbf{(C) }\frac{7}{27}}<math> |
~KingRavi | ~KingRavi | ||
== Solution 2 == | == Solution 2 == | ||
| − | We denote by <math>p< | + | We denote by </math>p<math> the fraction of red knights who are magical. |
Hence, | Hence, | ||
| Line 27: | Line 27: | ||
</cmath> | </cmath> | ||
| − | By solving this equation, we get <math>p = \frac{7}{27}< | + | By solving this equation, we get </math>p = \frac{7}{27}<math>. |
| − | Therefore, the answer is <math>\boxed{\textbf{(C) }\frac{7}{27}} | + | Therefore, the answer is </math>\boxed{\textbf{(C) }\frac{7}{27}}$. |
~Steven Chen (www.professorchenedu.com) | ~Steven Chen (www.professorchenedu.com) | ||
Revision as of 10:10, 17 August 2022
Problem
The knights in a certain kingdom come in two colors. 
of them are red, and the rest are blue. Furthermore, 
of the knights are magical, and the fraction of red knights who are magical is 
times the fraction of blue knights who are magical. What fraction of red knights are magical?
Solution 1
Let 
be the number of knights: then the number of red knights is 
and the number of blue knights is 
.
Let 
be the fraction of blue knights that are magical - then 
is the fraction of red knights that are magical. Thus we can write the equation 
We want to find the fraction of red knights that are magical, which is 
\boxed{\textbf{(C) }\frac{7}{27}}$~KingRavi
== Solution 2 == We denote by$ (Error compiling LaTeX. Unknown error_msg)p$the fraction of red knights who are magical.
Hence, <cmath> \[ \frac{1}{6} = \frac{2}{7} p + \left( 1 - \frac{2}{7} \right) \frac{p}{2} . \] </cmath>
By solving this equation, we get$ (Error compiling LaTeX. Unknown error_msg)p = \frac{7}{27}$.
Therefore, the answer is$ (Error compiling LaTeX. Unknown error_msg)\boxed{\textbf{(C) }\frac{7}{27}}$.
~Steven Chen (www.professorchenedu.com)
Video Solution by Interstigation
https://youtu.be/p9_RH4s-kBA?t=1274
Video Solution
~Education, the Study of Everything
See Also
| 2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
