Difference between revisions of "2008 AIME II Problems/Problem 2"
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== Problem == | == Problem == | ||
<!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Rudolph bikes at a [[constant]] rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the <math>50</math>-mile mark at exactly the same time. How many minutes has it taken them?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> | <!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Rudolph bikes at a [[constant]] rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the <math>50</math>-mile mark at exactly the same time. How many minutes has it taken them?<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> | ||
+ | |||
+ | == Simple Solution == | ||
+ | Let <math>r</math> be the time Rudolph takes disregarding breaks and <math>\frac{4}{3}r</math> be the time Jennifer takes disregarding breaks. We have the equation | ||
+ | <cmath>r+5\left(49\right)=\frac{4}{3}r+5\left(24\right)</cmath> | ||
+ | <cmath>123=\frac13r</cmath> | ||
+ | <cmath>r=375.</cmath> | ||
+ | Thus, the total time they take is <math>375 + 5(49) = \boxed{620}</math> minutes. | ||
== Solution == | == Solution == |
Revision as of 09:22, 18 July 2018
Contents
[hide]Problem
Rudolph bikes at a constant rate and stops for a five-minute break at the end of every mile. Jennifer bikes at a constant rate which is three-quarters the rate that Rudolph bikes, but Jennifer takes a five-minute break at the end of every two miles. Jennifer and Rudolph begin biking at the same time and arrive at the -mile mark at exactly the same time. How many minutes has it taken them?
Simple Solution
Let be the time Rudolph takes disregarding breaks and be the time Jennifer takes disregarding breaks. We have the equation Thus, the total time they take is minutes.
Solution
Let Rudolf bike at a rate , so Jennifer bikes at the rate . Let the time both take be .
Then Rudolf stops times (because the rest after he reaches the finish does not count), losing a total of minutes, while Jennifer stops times, losing a total of minutes. The time Rudolf and Jennifer actually take biking is then respectively.
Using the formula , since both Jennifer and Rudolf bike miles,
Substituting equation into equation and simplifying, we find
See also
2008 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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.