2021 Fall AMC 10B Problems/Problem 9
Contents
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
~KingRavi
Solution 2
We denote by the fraction of red knights who are magical.
Hence,
By solving this equation, we get .
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
Solution 3
Suppose there are knights. Then there are red knights and blue knights.
There are magical knights. Let be the number of red magical knights.
Then we have the equation . Solving, we get .
Plugging back gives us the fraction of red magical knights, which is .
~DairyQueenXD
Solution 4
Similar to Solution 3, let the total number of knights be (We can say this because the fractions we know have denominators of and which have a LCM of )
This means that there are red knights meaning there are blue knights
This also means that there are magical knights. This is important information for our next step.
We are given that the fraction of magical red knights is twice the fraction of magical blue knights. As this problem is dealing with the fraction being twice are large and is asking for the fraction, let's just deal with the variable being a fraction. Let be the fraction of red magical knights. This means that is the fraction of blue magical knights.
We know that total number of magical knights from what we figured out before is , so let's use the fraction we figured and make the equation:
Solving this gives us or the fraction of red magical knights.
Hence, the answer is
~ Batmanstark
Video Solution by Interstigation
https://youtu.be/p9_RH4s-kBA?t=1274
Video Solution
~Education, the Study of Everything
Video Solution by WhyMath
~savannahsolver
Video Solution by TheBeautyofMath
https://youtu.be/RyN-fKNtd3A?t=1190
~IceMatrix
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.