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Difference between revisions of "2023 AMC 8 Problems/Problem 20"

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== Problem ==
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Two integers are inserted into the list <math>3, 3, 8, 11, 28</math> to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
  
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<math>\textbf{(A) } 56 \qquad \textbf{(B) } 57 \qquad \textbf{(C) } 58 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 61</math>
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==Solution 1 ==
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To double the range, we must find the current range, which is <math>28 - 3 = 25</math>, to then double to: <math>2(25) = 50</math>. Since we do not want to change the median, we need to get a value less than <math>8</math> (as <math>8</math> would change the mode) for the smaller, making <math>53</math> fixed for the larger. Anything less than <math>3</math> is not beneficial to the optimization because you want to get the largest range without changing the mode. So, taking our optimal values of <math>7</math> and <math>53</math>, we have an answer of <math>7 + 53 = \boxed{\textbf{(D)}\ 60}</math>.
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~apex304, SohumUttamchandani, wuwang2002, TaeKim, CrystalFlower, CHECKMATE2021, leyele.lee
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==Video Solution by Math-X (Let's first Understand the question)==
 +
https://youtu.be/Ku_c1YHnLt0?si=1UUKWUVIPwomTx84&t=4411
 +
 +
~Math-X
 +
 +
==Video Solution (Solve under 60 seconds!!!)==
 +
https://youtu.be/6O5UXi-Jwv4?si=KvvABit-3-ZtX7Qa&t=925
 +
 +
~hsnacademy
 +
==Video Solution==
 +
https://youtu.be/BdkBQppueWY
 +
 +
Please like and subscribe
 +
 +
==Video Solution (CREATIVE THINKING(Very fast paced)!!!)==
 +
https://youtu.be/NpVLhU3AgNg
 +
 +
~Education, the Study of Everything
 +
 +
==Animated Video Solution==
 +
https://youtu.be/ItntB7vEafM
 +
 +
~Star League (https://starleague.us)
 +
 +
==Video Solution by OmegaLearn (Using Smart Sequence Analysis)==
 +
https://youtu.be/qNsgNa9Qq9M
 +
 +
==Video Solution by Magic Square==
 +
https://youtu.be/-N46BeEKaCQ?t=3136
 +
==Video Solution by Interstigation==
 +
https://youtu.be/DBqko2xATxs&t=2625
 +
 +
==Video Solution by WhyMath==
 +
https://youtu.be/lCVPFN1EK_M
 +
 +
~savannahsolver
 +
 +
==Video Solution by harungurcan==
 +
https://www.youtube.com/watch?v=Ki4tPSGAapU&t=534s
 +
 +
~harungurcan
 +
 +
==Video Solution by Dr. David==
 +
https://youtu.be/mMU-uvgwVzg
 +
 +
==See Also==
 +
{{AMC8 box|year=2023|num-b=19|num-a=21}}
 +
{{MAA Notice}}

Latest revision as of 14:36, 28 November 2024

Problem

Two integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?

$\textbf{(A) } 56 \qquad \textbf{(B) } 57 \qquad \textbf{(C) } 58 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 61$

Solution 1

To double the range, we must find the current range, which is $28 - 3 = 25$, to then double to: $2(25) = 50$. Since we do not want to change the median, we need to get a value less than $8$ (as $8$ would change the mode) for the smaller, making $53$ fixed for the larger. Anything less than $3$ is not beneficial to the optimization because you want to get the largest range without changing the mode. So, taking our optimal values of $7$ and $53$, we have an answer of $7 + 53 = \boxed{\textbf{(D)}\ 60}$.

~apex304, SohumUttamchandani, wuwang2002, TaeKim, CrystalFlower, CHECKMATE2021, leyele.lee

Video Solution by Math-X (Let's first Understand the question)

https://youtu.be/Ku_c1YHnLt0?si=1UUKWUVIPwomTx84&t=4411

~Math-X

Video Solution (Solve under 60 seconds!!!)

https://youtu.be/6O5UXi-Jwv4?si=KvvABit-3-ZtX7Qa&t=925

~hsnacademy

Video Solution

https://youtu.be/BdkBQppueWY

Please like and subscribe

Video Solution (CREATIVE THINKING(Very fast paced)!!!)

https://youtu.be/NpVLhU3AgNg

~Education, the Study of Everything

Animated Video Solution

https://youtu.be/ItntB7vEafM

~Star League (https://starleague.us)

Video Solution by OmegaLearn (Using Smart Sequence Analysis)

https://youtu.be/qNsgNa9Qq9M

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=3136

Video Solution by Interstigation

https://youtu.be/DBqko2xATxs&t=2625

Video Solution by WhyMath

https://youtu.be/lCVPFN1EK_M

~savannahsolver

Video Solution by harungurcan

https://www.youtube.com/watch?v=Ki4tPSGAapU&t=534s

~harungurcan

Video Solution by Dr. David

https://youtu.be/mMU-uvgwVzg

See Also

2023 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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