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Difference between revisions of "2024 AMC 8 Problems/Problem 6"

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==Problem==
 
==Problem==
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Sergai skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled P, Q, R, and S. What is the sorted order of the four paths from shortest to longest?
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[[Image:2024_AMC_8-Problem_6.png|500px]]
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<math>\textbf{(A)}\ P,Q,R,S \qquad \textbf{(B)}\ P,R,S,Q \qquad \textbf{(C)}\ Q,S,P,R \qquad \textbf{(D)}\ R,P,S,Q \qquad \textbf{(E)}\ R,S,P,Q</math>
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==Solution 1==
 
==Solution 1==
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You can measure the lengths of the paths until you find a couple of guaranteed true inferred statements as such:
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<math>Q</math> is greater than <math>S</math>,
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<math>P</math> is greater than <math>R</math>,
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and <math>R</math> and <math>P</math> are the smallest two, therefore the order is <math>R, P, S, Q.</math>
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Thus we get the answer <math>\boxed{\textbf{(D)}~R, P, S, Q}</math>.
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- U-King
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~ cxsmi (minor <math>\LaTeX</math> edits)
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~TabHawaii (minor formatting edits)
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==Solution 2 (Intuitive)==
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Obviously Path Q is the longest path, followed by Path S.
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So, it is down to Paths P and R.
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Notice that curved lines are always longer than the straight ones that meet their endpoints, therefore Path P is longer than Path R.
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Thus, the order from shortest to longest is <math>\boxed{\textbf{(D) } \text{R, P, S, Q}}</math>.
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~MrThinker
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==Video Solution by Math-X (First fully understand the problem!!!)==
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https://youtu.be/BaE00H2SHQM?si=ZedvqIYTDG3D20Rp&t=1301
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~Math-X
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==Video Solution by Power Solve (easy to digest!)==
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https://www.youtube.com/watch?v=16YYti_pDUg
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==Video Solution (A Clever Explanation You’ll Get Instantly)==
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https://youtu.be/5ZIFnqymdDQ?si=MmMWctYfzKIjwfE8&t=553
 +
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~hsnacademy
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==Video Solution 1 by NiuniuMaths (Easy to understand!)==
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https://www.youtube.com/watch?v=V-xN8Njd_Lc
 +
 +
~NiuniuMaths
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==Video Solution by Interstigation==
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https://youtu.be/ktzijuZtDas&t=386
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==Video Solution by Daily Dose of Math (Certified, Simple, and Logical)==
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https://youtu.be/WJCwG6olgNU
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~Thesmartgreekmathdude
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==Video Solution by WhyMath==
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https://youtu.be/mMDp6k_C6MI
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==See Also==
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{{AMC8 box|year=2024|num-b=5|num-a=7}}
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{{MAA Notice}}

Latest revision as of 15:50, 20 November 2024

Problem

Sergai skated around an ice rink, gliding along different paths. The gray lines in the figures below show four of the paths labeled P, Q, R, and S. What is the sorted order of the four paths from shortest to longest?

2024 AMC 8-Problem 6.png

$\textbf{(A)}\ P,Q,R,S \qquad \textbf{(B)}\ P,R,S,Q \qquad \textbf{(C)}\ Q,S,P,R \qquad \textbf{(D)}\ R,P,S,Q \qquad \textbf{(E)}\ R,S,P,Q$

Solution 1

You can measure the lengths of the paths until you find a couple of guaranteed true inferred statements as such: $Q$ is greater than $S$, $P$ is greater than $R$, and $R$ and $P$ are the smallest two, therefore the order is $R, P, S, Q.$ Thus we get the answer $\boxed{\textbf{(D)}~R, P, S, Q}$.

- U-King

~ cxsmi (minor $\LaTeX$ edits)

~TabHawaii (minor formatting edits)

Solution 2 (Intuitive)

Obviously Path Q is the longest path, followed by Path S.

So, it is down to Paths P and R.

Notice that curved lines are always longer than the straight ones that meet their endpoints, therefore Path P is longer than Path R.

Thus, the order from shortest to longest is $\boxed{\textbf{(D) } \text{R, P, S, Q}}$.

~MrThinker

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

https://youtu.be/BaE00H2SHQM?si=ZedvqIYTDG3D20Rp&t=1301

~Math-X

Video Solution by Power Solve (easy to digest!)

https://www.youtube.com/watch?v=16YYti_pDUg

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/5ZIFnqymdDQ?si=MmMWctYfzKIjwfE8&t=553

~hsnacademy

Video Solution 1 by NiuniuMaths (Easy to understand!)

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

~NiuniuMaths

Video Solution by Interstigation

https://youtu.be/ktzijuZtDas&t=386

Video Solution by Daily Dose of Math (Certified, Simple, and Logical)

https://youtu.be/WJCwG6olgNU

~Thesmartgreekmathdude

Video Solution by WhyMath

https://youtu.be/mMDp6k_C6MI

See Also

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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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