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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?
  
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== Solution One ==
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== Solution ==
  
 
Let <math>r</math> represent the rate Tom swims in miles per minute. Then we have
 
Let <math>r</math> represent the rate Tom swims in miles per minute. Then we have
  
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<math>\frac{.5}{r} + \frac{8}{5r} + \frac{30}{10r} = 255</math>
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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>
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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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== Solution Two ==
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==Video Solution==
  
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Tom trips and dies, therefore, because the decomposition of his body is similar to the fractional decomposition of the formula <math>\frac{.5}{r} + \frac{8}{5r} + \frac{30}{10r} = 255</math>, we can easily express it in terms of e^i<math>\pi</math>theta, through Euler's formula. i is <math>\sqrt-1</math>, but we were taught in Algebra class not to worry about the <math>\sqrt  </math> of negative numbers, therefore the answer is <math>\boxed{000}</math>
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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 19:03, 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 (Problems • Answer Key • Resources)
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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