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2023 AMC 8 Problems/Problem 20

Revision as of 21:32, 24 January 2023 by Pi is 3.14 (talk | contribs)

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?

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

Solutions

We have the set {3, 3, 8, 16, 28}. To double the range we must find the current range, which is $28 - 3 = 25$, to then the double is $2(25) = 50$ Since we dont want to change the median we need to get a value greater than 8 (as 8 would change the mode) for the smaller and 53 is fixed for the larger as anything less than 3 is not beneficial to the optimization. So taking our optimal values of 53 and 7 we have an answer of $53 + 7 = \boxed{\text{(D)}60}$ -apex304, SohumUttamchandani, wuwang2002, TaeKim

Animated Video Solution

https://youtu.be/ItntB7vEafM

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Video Solution by OmegaLearn (Using Smart Sequence Analysis)

https://youtu.be/qNsgNa9Qq9M

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