Difference between revisions of "2009 AMC 10A Problems/Problem 20"
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==Solution 2== | ==Solution 2== | ||
| − | Because the speed of Andrea is 3 times as fast as Lauren and the distance between them is decreasing at a rate of 1 kilometer per minute, Andrea's speed is <math>\frac{3}{4} \textbf{km/min}</math>, and Lauren's <math>\frac{1}{4} \textbf{km/min}</math>. Therefore, after 5 minutes, Andrea will have biked <math>\frac{3}{4} \cdot 5 = \frac{15}{4}</math> | + | Because the speed of Andrea is 3 times as fast as Lauren and the distance between them is decreasing at a rate of 1 kilometer per minute, Andrea's speed is <math>\frac{3}{4} \textbf{km/min}</math>, and Lauren's <math>\frac{1}{4} \textbf{km/min}</math>. Therefore, after 5 minutes, Andrea will have biked <math>\frac{3}{4} \cdot 5 = \frac{15}{4}km</math>. |
In all, Lauren will have to bike <math>20 - \frac{15}{4} = \frac{80}{4} - \frac{15}{4} = \frac{65}{4}</math>km. Because her speed is <math>\frac{1}{4} \textbf{km/min}</math>, the time elapsed will be <math>\frac{\frac{65}{4}}{\frac{1}{4}} = \boxed{\textbf{D) 65}}</math> | In all, Lauren will have to bike <math>20 - \frac{15}{4} = \frac{80}{4} - \frac{15}{4} = \frac{65}{4}</math>km. Because her speed is <math>\frac{1}{4} \textbf{km/min}</math>, the time elapsed will be <math>\frac{\frac{65}{4}}{\frac{1}{4}} = \boxed{\textbf{D) 65}}</math> | ||
Revision as of 08:59, 8 June 2021
Problem
Andrea and Lauren are 
kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of 
kilometer per minute. After 
minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?
Solution 1
Let their speeds in kilometers per hour be 
and 
. We know that 
and that 
. (The second equation follows from the fact that 
.) This solves to 
and 
.
As the distance decreases at a rate of kilometer per minute, after minutes the distance between them will be 
kilometers.
From this point on, only Lauren will be riding her bike. As there are 
kilometers remaining and , she will need exactly an hour to get to Andrea. Therefore the total time in minutes is 
.
Solution 2
Because the speed of Andrea is 3 times as fast as Lauren and the distance between them is decreasing at a rate of 1 kilometer per minute, Andrea's speed is 
, and Lauren's 
. Therefore, after 5 minutes, Andrea will have biked 
.
In all, Lauren will have to bike 
km. Because her speed is , the time elapsed will be 
Video Solution
~savannahsolver
See Also
| 2009 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
