2023 AMC 8 Problems/Problem 13: Difference between revisions
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==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 3rd water station is located 2 miles after the 1st repair station. How long is the race | |||
in miles? | |||
== | [[File:2023 AMC 8-13.png|thumb|center|300px]] | ||
<math>\textbf{(A)}\ 8 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 96</math> | |||
==Solution== | |||
Knowing that there are <math>7</math> equally spaced water stations they are each located <math>\frac{d}{8}</math>, <math>\frac{2d}{8}</math>,… <math>\frac{7d}{8}</math> of the way from the start. Using the same logic for the <math>3</math> station we have <math>\frac{d}{3}</math> and <math>\frac{2d}{3}</math> for the repair stations. It is given that the 3rd water is <math>2</math> miles ahead of the <math>1</math>st repair station. So setting an equation we have <math>\frac{3d}{8} = \frac{d}{3} + 2</math> getting common denominators <math>\frac{9d}{24} = \frac{8d}{24} + 2</math> so then we have <math>d = \boxed{\text{(D)}48}</math> from this. | Knowing that there are <math>7</math> equally spaced water stations they are each located <math>\frac{d}{8}</math>, <math>\frac{2d}{8}</math>,… <math>\frac{7d}{8}</math> of the way from the start. Using the same logic for the <math>3</math> station we have <math>\frac{d}{3}</math> and <math>\frac{2d}{3}</math> for the repair stations. It is given that the 3rd water is <math>2</math> miles ahead of the <math>1</math>st repair station. So setting an equation we have <math>\frac{3d}{8} = \frac{d}{3} + 2</math> getting common denominators <math>\frac{9d}{24} = \frac{8d}{24} + 2</math> so then we have <math>d = \boxed{\text{(D)}48}</math> from this. | ||
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== | ==Video Solution (Animated)== | ||
https://youtu.be/NivfOThj1No | https://youtu.be/NivfOThj1No | ||
~Star League (https://starleague.us) | ~Star League (https://starleague.us) | ||
Revision as of 04:52, 25 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 3rd water station is located 2 miles after the 1st repair station. How long is the race in miles?
Solution
Knowing that there are 
equally spaced water stations they are each located 
, 
,… 
of the way from the start. Using the same logic for the 
station we have 
and 
for the repair stations. It is given that the 3rd water is 
miles ahead of the 
st repair station. So setting an equation we have 
getting common denominators 
so then we have 
from this.
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat
Video Solution (Animated)
~Star League (https://starleague.us)
