Difference between revisions of "2023 AMC 8 Problems/Problem 13"
Torpedorocks (talk | contribs) (→Solution) |
MRENTHUSIASM (talk | contribs) |
||
| Line 32: | Line 32: | ||
</asy> | </asy> | ||
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96</math> | <math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96</math> | ||
| + | |||
| + | ==Solution== | ||
| + | |||
| + | Suppose that the race is <math>d</math> miles long. The water stations are located at <cmath>\frac{d}{8}, \frac{2d}{8}, \ldots, \frac{7d}{8}</cmath> miles from the start, and the repair stations are located at <cmath>\frac{d}{3}, \frac{2d}{3}</cmath> miles from the start. | ||
| + | |||
| + | We are given that <math>\frac{3d}{8}=\frac{d}{3}+2,</math> from which | ||
| + | <cmath>\begin{align*} | ||
| + | \frac{9d}{24}&=\frac{8d}{24}+2 \\ | ||
| + | \frac{d}{24}&=2 \\ | ||
| + | d&=\boxed{\textbf{(D)}\ 48}. | ||
| + | \end{align*}</cmath> | ||
| + | ~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat, MRENTHUSIASM | ||
==Video Solution by Math-X (Let's first Understand the question)== | ==Video Solution by Math-X (Let's first Understand the question)== | ||
Revision as of 01:34, 7 January 2024
Contents
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 (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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
