Difference between revisions of "2021 AMC 12B Problems/Problem 3"
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− | + | ==Problem== | |
+ | Suppose<cmath>2+\frac{1}{1+\frac{1}{2+\frac{2}{3+x}}}=\frac{144}{53}.</cmath>What is the value of <math>x?</math> | ||
+ | |||
+ | <math>\textbf{(A) }\frac34 \qquad \textbf{(B) }\frac78 \qquad \textbf{(C) }\frac{14}{15} \qquad \textbf{(D) }\frac{37}{38} \qquad \textbf{(E) }\frac{52}{53}</math> | ||
+ | |||
+ | ==Solution== | ||
+ | Subtracting <math>2</math> from both sides and taking reciprocals gives <math>1+\frac{1}{2+\frac{2}{3+x}}=\frac{53}{38}</math>. Subtracting <math>1</math> from both sides and taking reciprocals again gives <math>2+\frac{2}{3+x}=\frac{38}{15}</math>. Subtracting <math>2</math> from both sides and taking reciprocals for the final time gives <math>\frac{x+3}{2}=\frac{15}{8}</math> or <math>x=\frac{3}{4} \implies \boxed{\text{A}}</math>. | ||
+ | |||
+ | ~ OlympusHero | ||
+ | |||
+ | ==Video Solution by Punxsutawney Phil== | ||
+ | https://youtube.com/watch?v=qpvS2PVkI8A&t=127s | ||
+ | |||
+ | == Video Solution by OmegaLearn (Algebraic Manipulations) == | ||
+ | https://youtu.be/WskJI8_7Gk0 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | ==Video Solution by Hawk Math== | ||
+ | https://www.youtube.com/watch?v=VzwxbsuSQ80 | ||
+ | |||
+ | ==Video Solution by TheBeautyofMath== | ||
+ | https://youtu.be/EMzdnr1nZcE?t=384 | ||
+ | |||
+ | ~IceMatrix | ||
+ | |||
+ | ==Video Solution (Just 1 min!)== | ||
+ | https://youtu.be/UhoVUqZjfhg | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC12 box|year=2021|ab=B|num-b=2|num-a=4}} | ||
+ | {{MAA Notice}} |
Latest revision as of 23:50, 18 July 2023
Contents
Problem
SupposeWhat is the value of
Solution
Subtracting from both sides and taking reciprocals gives
. Subtracting
from both sides and taking reciprocals again gives
. Subtracting
from both sides and taking reciprocals for the final time gives
or
.
~ OlympusHero
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=qpvS2PVkI8A&t=127s
Video Solution by OmegaLearn (Algebraic Manipulations)
~ pi_is_3.14
Video Solution by Hawk Math
https://www.youtube.com/watch?v=VzwxbsuSQ80
Video Solution by TheBeautyofMath
https://youtu.be/EMzdnr1nZcE?t=384
~IceMatrix
Video Solution (Just 1 min!)
~Education, the Study of Everything
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 2 |
Followed by Problem 4 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.