Difference between revisions of "2016 AMC 8 Problems/Problem 16"
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| + | ==Problem== | ||
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Annie and Bonnie are running laps around a <math>400</math>-meter oval track. They started together, but Annie has pulled ahead, because she runs <math>25\%</math> faster than Bonnie. How many laps will Annie have run when she first passes Bonnie? | Annie and Bonnie are running laps around a <math>400</math>-meter oval track. They started together, but Annie has pulled ahead, because she runs <math>25\%</math> faster than Bonnie. How many laps will Annie have run when she first passes Bonnie? | ||
<math>\textbf{(A) }1\dfrac{1}{4}\qquad\textbf{(B) }3\dfrac{1}{3}\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }25</math> | <math>\textbf{(A) }1\dfrac{1}{4}\qquad\textbf{(B) }3\dfrac{1}{3}\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }25</math> | ||
| − | ==Solution== | + | ==Solutions== |
| − | Each lap Bonnie runs, Annie runs another quarter lap, so Bonnie will run four laps before she is overtaken. This means that Annie and Bonnie are equal so that Annie needs to run another lap to overtake Bonnie. That means Annie will have run <math>\boxed{\textbf{(D)}\ 5 }</math> laps. | + | ===Solution 1=== |
| − | + | Each lap Bonnie runs, Annie runs another quarter lap, so Bonnie will run four laps before she is overtaken. This means that Annie and Bonnie are equal so that Annie needs to run another lap to overtake Bonnie. That means Annie will have run <math>\boxed{\textbf{(D)}\ 5 }</math> laps. | |
| − | {{ | + | |
| + | ===Solution 2=== | ||
| + | Saying that Bonnie runs 40 meters per second, then Annie will run 50 meters per second. It will take Bonnie 10 seconds to run 1 lap and it would take 8 seconds for Annie to do 1 lap. The LCM of 8 and 10 is 40. In 40 seconds Bonnie will do 4 laps and Annie will do 5 laps. Since Annie did 1 more lap she passed Bonnie. So the answer is <math>\boxed{\textbf{(D)}\ 5 }</math> laps. | ||
| + | |||
| + | == Video Solution by Pi Academy == | ||
| + | |||
| + | https://youtu.be/Sn1QLgCJUAQ?si=UebITjxagXo7KoDI | ||
| + | |||
| − | ==Solution 2== | + | ==Video Solution 2== |
| − | + | https://youtu.be/lRbxzdBZpIY | |
| − | + | ~savannahsolver | |
| − | + | == See Also == | |
| + | {{AMC8 box|year=2016|num-b=15|num-a=17}} | ||
| + | {{MAA Notice}} | ||
| + | [[Category:Introductory Algebra Problems]] | ||
Latest revision as of 14:12, 28 July 2025
Contents
Problem
Annie and Bonnie are running laps around a 
-meter oval track. They started together, but Annie has pulled ahead, because she runs 
faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?
Solutions
Solution 1
Each lap Bonnie runs, Annie runs another quarter lap, so Bonnie will run four laps before she is overtaken. This means that Annie and Bonnie are equal so that Annie needs to run another lap to overtake Bonnie. That means Annie will have run 
laps.
Solution 2
Saying that Bonnie runs 40 meters per second, then Annie will run 50 meters per second. It will take Bonnie 10 seconds to run 1 lap and it would take 8 seconds for Annie to do 1 lap. The LCM of 8 and 10 is 40. In 40 seconds Bonnie will do 4 laps and Annie will do 5 laps. Since Annie did 1 more lap she passed Bonnie. So the answer is laps.
Video Solution by Pi Academy
https://youtu.be/Sn1QLgCJUAQ?si=UebITjxagXo7KoDI
Video Solution 2
~savannahsolver
See Also
| 2016 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
