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Difference between revisions of "2023 AMC 8 Problems/Problem 13"

(Asy replaces png.)
Line 5: Line 5:
 
finish lines. The <math>3</math>rd water station is located <math>2</math> miles after the <math>1</math>st repair station. How long is the race
 
finish lines. The <math>3</math>rd water station is located <math>2</math> miles after the <math>1</math>st repair station. How long is the race
 
in miles?
 
in miles?
−
 
 
<asy>
 
<asy>
−
//Diagram By TheMathGuyd, The road is widened for bicycle safety
+
usepackage("mathptmx");
 
size(10cm);
 
size(10cm);
−
real tl = 0.2;
+
filldraw((11,4.5)--(171,4.5)--(171,17.5)--(11,17.5)--cycle,mediumgray*0.4 + lightgray*0.6);
−
real dpth = 0.8;
+
draw((11,11)--(171,11),linetype("2 2")+white+linewidth(1.2));
−
fill((0,0)--(8,0)--(8,1)--(0,1)--cycle,mediumgrey);
+
draw((0,0)--(11,0)--(11,22)--(0,22)--cycle);
−
draw((0,.5)--(8,.5),dashed+white);
+
draw((171,0)--(182,0)--(182,22)--(171,22)--cycle);
−
draw((0,0)--(8,0)--(8,1)--(0,1)--cycle,black);
+
 
−
draw((1,0)--(1,-tl));
+
draw((31,4.5)--(31,0));
−
draw((2,0)--(2,-tl));
+
draw((51,4.5)--(51,0));
−
draw((7,0)--(7,-tl));
+
draw((151,4.5)--(151,0));
−
label(rotate(45)*"Water 1",(0.6,-dpth));
+
 
−
label(rotate(45)*"Water 2",(1.6,-dpth));
+
label(scale(.85)*rotate(45)*"Water 1", (23,-13.5));
−
label(rotate(45)*"Water 7",(6.6,-dpth));
+
label(scale(.85)*rotate(45)*"Water 2", (43,-13.5));
−
label("\ldots\ldots",(4.2,-dpth));
+
label(scale(.85)*rotate(45)*"Water 7", (143,-13.5));
−
label(rotate(90)*"Start",(0,0.5),W);
+
 
−
label(rotate(-90)*"Finish",(8,0.5),E);
+
filldraw(circle((103,-13.5),.2));
−
//Someone is going to see this and find out I don't know how to put borders on text and laugh.
+
filldraw(circle((98,-13.5),.2));
−
real lh = 0.45;
+
filldraw(circle((93,-13.5),.2));
−
real lr = 0.1;
+
filldraw(circle((88,-13.5),.2));
−
path bx = (-lh,-lr)--(0,-lr)--(0,1+lr)--(-lh,1+lr)--cycle;
+
filldraw(circle((83,-13.5),.2));
−
draw(bx);
+
 
−
draw(reflect((4,0),(4,1))*bx);
+
label(scale(.85)*rotate(90)*"Start", (5.5,11));
 +
label(scale(.85)*rotate(270)*"Finish", (176.5,11));
 
</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>
  

Revision as of 22:48, 26 January 2023

Problem

Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles? [asy] 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] $\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96$

Solution

Suppose that the race is $d$ miles long. The water stations are located at \[\frac{d}{8}, \frac{2d}{8}, \ldots, \frac{7d}{8}\] miles from the start, and the repair stations are located at \[\frac{d}{3}, \frac{2d}{3}\] miles from the start.

We are given that $\frac{3d}{8}=\frac{d}{3}+2,$ from which \begin{align*} \frac{9d}{24}&=\frac{8d}{24}+2 \\ \frac{d}{24}&=2 \\ d&=\boxed{\textbf{(D)}\ 48}. \end{align*} ~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat, MRENTHUSIASM

Video Solution (Animated)

https://youtu.be/NivfOThj1No

~Star League (https://starleague.us)

Video Solution by Magic Square

https://youtu.be/-N46BeEKaCQ?t=4439

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. AMC logo.png

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