Difference between revisions of "2021 AMC 10A Problems/Problem 6"
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==Solution 2 (Convenient Distance)== | ==Solution 2 (Convenient Distance)== | ||
| − | <i><b>We will follow the same template as shown in Solution 1, except that we will replace <math>\boldsymbol{d}</math> with a | + | <i><b>We will follow the same template as shown in Solution 1, except that we will replace <math>\boldsymbol{d}</math> with a convenient constant.</b></i> |
Let <math>24</math> miles be the distance from the trailhead to the fire tower. When Chantal meets Jean, the two have traveled for <cmath>\frac{12}{4} + \frac{12}{2}+\frac{12}{3}=3+6+4=13</cmath> hours. At that point, Jean has traveled for <math>12</math> miles, so his average speed is <math>\boxed{\textbf{(A)} ~\frac{12}{13}}</math> miles per hour. | Let <math>24</math> miles be the distance from the trailhead to the fire tower. When Chantal meets Jean, the two have traveled for <cmath>\frac{12}{4} + \frac{12}{2}+\frac{12}{3}=3+6+4=13</cmath> hours. At that point, Jean has traveled for <math>12</math> miles, so his average speed is <math>\boxed{\textbf{(A)} ~\frac{12}{13}}</math> miles per hour. | ||
Revision as of 21:24, 26 May 2021
Contents
Problem
Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at 
miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to 
miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at 
miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?
Solution 1 (Generalized Distance)
Let 
miles be the distance from the trailhead to the fire tower, where 
When Chantal meets Jean, the two have traveled for ![\[\frac d4 + \frac d2 + \frac d3 = d\left(\frac 14 + \frac 12 + \frac 13\right) =d\left(\frac{3}{12} + \frac{6}{12} + \frac{4}{12}\right)=\frac{13}{12}d\]](http://latex.artofproblemsolving.com/c/0/a/c0a9fe24b85132a945c1f2f286ed5334c47cc127.png)
hours. At that point, Jean has traveled for 
miles, so his average speed is 
miles per hour.
~MRENTHUSIASM
Solution 2 (Convenient Distance)
We will follow the same template as shown in Solution 1, except that we will replace 
with a convenient constant.
Let 
miles be the distance from the trailhead to the fire tower. When Chantal meets Jean, the two have traveled for ![\[\frac{12}{4} + \frac{12}{2}+\frac{12}{3}=3+6+4=13\]](http://latex.artofproblemsolving.com/8/a/6/8a64535fec40239f4279ca8fb392c586b0977eaf.png)
hours. At that point, Jean has traveled for 
miles, so his average speed is 
miles per hour.
~MRENTHUSIASM
Video Solution 1 (Using Speed, Time, Distance)
~ pi_is_3.14
Video Solution 2 (Simple and Quick)
~ Education, the Study of Everything
Video Solution 3
~savannahsolver
Video Solution 4 (by TheBeautyofMath)
~IceMatrix
See Also
| 2021 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 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. 
