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

Revision as of 09:49, 23 February 2023

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

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. Remember, anything less than $3$ is not beneficial to the optimization. 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

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/1bA7fD7Lg54?t=1970

Video Solution by WhyMath

https://youtu.be/lCVPFN1EK_M

~savannahsolver

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.

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 ==Solution==
-To double the range we must find the current range, which is <math>28 - 3 = 25</math>, to then the double is <math>2(25) = 50</math> Since we dont want to change the median we need to get a value greater than <math>8</math> (as <math>8</math> would change the mode) for the smaller and <math>53</math> is fixed for the larger as anything less than <math>3</math> is not beneficial to the optimization. So taking our optimal values of <math>53</math> and <math>7</math> we have an answer of <math>53 + 7 = \boxed{\textbf{(D)}\ 60}</math>.
+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. Remember, anything less than <math>3</math> is not beneficial to the optimization. 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>.
 ~apex304, SohumUttamchandani, wuwang2002, TaeKim, CrystalFlower

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