2023 AMC 8 Problems/Problem 13: Difference between revisions
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==Video Solution by Math-X (Let's first Understand the question)== | ==Video Solution by Math-X (Let's first Understand the question)== | ||
https://youtu.be/Ku_c1YHnLt0?si=YRjrl2U0waLkNWqm&t=2151 ~MATH-X | https://youtu.be/Ku_c1YHnLt0?si=YRjrl2U0waLkNWqm&t=2151 ~MATH-X | ||
==Video Solution (Solve under 60 seconds!!!)== | |||
https://youtu.be/6O5UXi-Jwv4?si=KvvABit-3-ZtX7Qa&t=588 | |||
~hsnacademy | |||
==Video Solution (CREATIVE THINKING!!!)== | ==Video Solution (CREATIVE THINKING!!!)== | ||
Revision as of 01:11, 21 August 2024
Problem
Along the route of a bicycle race, 
water stations are evenly spaced between the start and finish lines,
as shown in the figure below. There are also 
repair stations evenly spaced between the start and
finish lines. The 
rd water station is located miles after the 
st repair station. How long is the race
in miles?
![[asy] // Credits given to Themathguyd and Kante314 usepackage("mathptmx"); size(10cm); filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray*0.4 + lightgray*0.6); draw((11,11)--(171,11),linetype("2 2")+white+linewidth(1.2)); draw((0,0)--(11,0)--(11,22)--(0,22)--cycle); draw((171,0)--(182,0)--(182,22)--(171,22)--cycle); draw((31,4.5)--(31,0)); draw((51,4.5)--(51,0)); draw((151,4.5)--(151,0)); label(scale(.85)*rotate(45)*"Water 1", (23,-13.5)); label(scale(.85)*rotate(45)*"Water 2", (43,-13.5)); label(scale(.85)*rotate(45)*"Water 7", (143,-13.5)); filldraw(circle((103,-13.5),.2)); filldraw(circle((98,-13.5),.2)); filldraw(circle((93,-13.5),.2)); filldraw(circle((88,-13.5),.2)); filldraw(circle((83,-13.5),.2)); label(scale(.85)*rotate(90)*"Start", (5.5,11)); label(scale(.85)*rotate(270)*"Finish", (176.5,11)); [/asy]](http://latex.artofproblemsolving.com/e/f/a/efaa138f2f7bc240bc49299009af49a7fc9a0420.png)
Solution
Suppose that the race is 
miles long. The water stations are located at ![\[\frac{d}{8}, \frac{2d}{8}, \ldots, \frac{7d}{8}\]](http://latex.artofproblemsolving.com/5/d/7/5d75992c71e35e3d504ae9cbb3b9aeeb572d0fbb.png)
miles from the start, and the repair stations are located at ![\[\frac{d}{3}, \frac{2d}{3}\]](http://latex.artofproblemsolving.com/b/d/c/bdc9e9aa2443a5c15309d91a1a860ab82d30094a.png)
miles from the start.
We are given that 
from which
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat, MRENTHUSIASM
Video Solution by Math-X (Let's first Understand the question)
https://youtu.be/Ku_c1YHnLt0?si=YRjrl2U0waLkNWqm&t=2151 ~MATH-X
Video Solution (Solve under 60 seconds!!!)
https://youtu.be/6O5UXi-Jwv4?si=KvvABit-3-ZtX7Qa&t=588
~hsnacademy
Video Solution (CREATIVE THINKING!!!)
~Education, the Study of Everything
Video Solution (Animated)
~Star League (https://starleague.us)
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=4439
Video Solution by Interstigation
https://youtu.be/DBqko2xATxs&t=1299
Video Solution by harungurcan
https://www.youtube.com/watch?v=VqN7c5U5o98&t=16s
~harungurcan
See Also
| 2023 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination
