Difference between revisions of "2013 AIME I Problems/Problem 1"
m (added the problem and basic layout) |
Hashtagmath (talk | contribs) |
||
(17 intermediate revisions by 12 users not shown) | |||
Line 1: | Line 1: | ||
− | == Problem | + | == 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 == | ||
− | ( | + | Let <math>r</math> represent the rate Tom swims in miles per minute. Then we have |
+ | |||
+ | <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. | ||
+ | |||
+ | ==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}} |
Latest revision as of 20:03, 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.