Difference between revisions of "2013 AIME I Problems/Problem 1"

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== Problem 1 ==
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== Problem ==
 
The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
 
The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
  
 
== Solution ==
 
== Solution ==
  
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Let <math>r</math> represent the rate Tom swims in miles per minute. Then we have
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<math>\frac{1/2}{r} + \frac{8}{5r} + \frac{30}{10r} = 255</math>
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Solving for <math>r</math>, we find <math>r = 1/50</math>, so the time Tom spends biking is <math>\frac{30}{(10)(1/50)} = \boxed{150}</math> minutes.
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==Video Solution==
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https://www.youtube.com/watch?v=kz3ZX4PT-_0
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~Shreyas S
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2013|n=I|before=First Problem|num-a=2}}
 
{{AIME box|year=2013|n=I|before=First Problem|num-a=2}}
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{{MAA Notice}}

Latest revision as of 20:03, 24 January 2021

Problem

The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

Solution

Let $r$ represent the rate Tom swims in miles per minute. Then we have

$\frac{1/2}{r} + \frac{8}{5r} + \frac{30}{10r} = 255$

Solving for $r$, we find $r = 1/50$, so the time Tom spends biking is $\frac{30}{(10)(1/50)} = \boxed{150}$ minutes.

Video Solution

https://www.youtube.com/watch?v=kz3ZX4PT-_0 ~Shreyas S

See also

2013 AIME I (ProblemsAnswer KeyResources)
Preceded by
First Problem
Followed by
Problem 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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