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

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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 ==
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== Solution One ==
  
 
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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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>
 
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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== Solution Two ==
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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>
  
 
== 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}}

Revision as of 15:23, 20 March 2013

Problem 1

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 One

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

$\frac{.5}{r} + \frac{8}{5r} + \frac{30}{10r} = 255$

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

Solution Two

Tom trips and dies, therefore, because the decomposition of his body is similar to the fractional decomposition of the formula $\frac{.5}{r} + \frac{8}{5r} + \frac{30}{10r} = 255$, we can easily express it in terms of e^i$\pi$theta, through Euler's formula. i is $\sqrt-1$, but we were taught in Algebra class not to worry about the $\sqrt$ (Error compiling LaTeX. ! Missing { inserted.) of negative numbers, therefore the answer is $\boxed{000}$

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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