Difference between revisions of "2013 AIME I Problems/Problem 1"
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| − | + | == Problem == | |
| − | ==Problem | ||
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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? | ||
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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 | ||
| − | <math>\frac{ | + | <math>\frac{1/2}{r} + \frac{8}{5r} + \frac{30}{10r} = 255</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> minutes. | 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== | ||
| + | |||
| + | https://www.youtube.com/watch?v=kz3ZX4PT-_0 | ||
| + | ~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}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 19:03, 24 January 2021
Contents
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 
represent the rate Tom swims in miles per minute. Then we have
Solving for , we find 
, so the time Tom spends biking is 
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
