Difference between revisions of "2024 AMC 8 Problems/Problem 14"

Revision as of 08:04, 31 January 2024

Problem

The one-way routes connecting towns $A,M,C,X,Y,$ and $Z$ are shown in the figure below(not drawn to scale).The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance form A to Z in kilometers?

$\textbf{(A)}\ 28 \qquad \textbf{(B)}\ 29 \qquad \textbf{(C)}\ 30 \qquad \textbf{(D)}\ 31 \qquad \textbf{(E)}\ 32$

Solution 1

We can simply see that path $A \rightarrow X \rightarrow M \rightarrow Y \rightarrow C \rightarrow Z$ will give us the smallest value. Adding, $5+2+6+5+10 = \boxed{28}$ . This is nice as it’s also the smallest value, solidifying our answer.

You can also simply brute-force it or sort of think ahead - for example, getting from A to M can be done $2$ ways; $A \rightarrow X \rightarrow M$ ( $5+2$ ) or $A \rightarrow M (8)$ , so you should take the shorter route ( $5+2$ ). Another example is M to C, two ways - one is $6+5$ and the other is $14$ . Take the shorter route. After this, you need to consider a few more times - consider if $5+10$ ( $Y \rightarrow C \rightarrow Z$ ) is greater than $17 (Y \rightarrow Z$ ), which it is not, and consider if $25 (M \rightarrow Z$ ) is greater than $14+10$ ( $M \rightarrow C \rightarrow Z$ ) or $6+5+10$ ( $M \rightarrow Y \rightarrow C \rightarrow Z$ ) which it is not. TL;DR: $5+2=6=5=10 = \boxed{28}$ . [Note: This is probably just the thinking behind the solution.] {Double-note: As MaxyMoosy said, since this answer is the smallest one, it has to be the right answer.}

~MaxyMoosy ~HACKER2022

Video Solution 1 (easy to digest) by Power Solve

https://youtu.be/2AVrraozwF8

Video Solution 2 by SpreadTheMathLove

https://www.youtube.com/watch?v=RRTxlduaDs8

Video Solution 3 by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=V-xN8Njd_Lc

~NiuniuMaths

Video Solution by CosineMethod [🔥Fast and Easy🔥]

https://www.youtube.com/watch?v=bAxLRYT6SCw

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/BaE00H2SHQM?si=y_YVZFC4DQdB2qhM&t=3280

~Math-X

@@ Line 27: / Line 27: @@
 https://www.youtube.com/watch?v=bAxLRYT6SCw
+==Video Solution by Math-X (First fully understand the problem!!!)==
+https://youtu.be/BaE00H2SHQM?si=y_YVZFC4DQdB2qhM&t=3280
+~Math-X
 ==See Also==
 {{AMC8 box|year=2024|num-b=13|num-a=15}}
 {{MAA Notice}}

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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

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