Difference between revisions of "2019 AMC 8 Problems/Problem 16"
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==Solution 3== | ==Solution 3== | ||
If he travels <math>15</math> miles at a speed of <math>30</math> miles per hour, he travels for 30 min. Average rate is total distance over total time so <math>(15+d)/(0.5 + t) = 50</math>, where d is the distance left to travel and t is the time to travel that distance. Solve for <math>d</math> to get <math>d = 10+50t</math>. You also know that he has to travel <math>55</math> miles per hour for some time, so <math>d=55t</math>. Plug that in for d to get <math>55t = 10+50t</math> and <math>t=2</math> and since <math>d=55t</math>, <math>d = 2\cdot55 =110</math>, the answer is <math>\boxed{\textbf{(D)}\ 110}</math>. | If he travels <math>15</math> miles at a speed of <math>30</math> miles per hour, he travels for 30 min. Average rate is total distance over total time so <math>(15+d)/(0.5 + t) = 50</math>, where d is the distance left to travel and t is the time to travel that distance. Solve for <math>d</math> to get <math>d = 10+50t</math>. You also know that he has to travel <math>55</math> miles per hour for some time, so <math>d=55t</math>. Plug that in for d to get <math>55t = 10+50t</math> and <math>t=2</math> and since <math>d=55t</math>, <math>d = 2\cdot55 =110</math>, the answer is <math>\boxed{\textbf{(D)}\ 110}</math>. | ||
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==Video Solution== | ==Video Solution== | ||
Revision as of 16:04, 26 December 2022
Contents
Problem 16
Qiang drives 
miles at an average speed of 
miles per hour. How many additional miles will he have to drive at 
miles per hour to average 
miles per hour for the entire trip?
Solution 1
The only option that is easily divisible by is 
, which gives 2 hours of travel. And, the formula is 
.
And, 
= 
.
Thus, 
.
Both are equal and thus our answer is 
Solution 2
To calculate the average speed, simply evaluate the total distance over the total time. Let the number of additional miles he has to drive be 
Therefore, the total distance is 
and the total time (in hours) is ![\[\frac{15}{30}+\frac{x}{55}=\frac{1}{2}+\frac{x}{55}.\]](http://latex.artofproblemsolving.com/8/f/a/8fa0a8f8d7aee64b52fba8f7ec3c7cc0e5a67416.png)
We can set up the following equation: ![\[\frac{15+x}{\frac{1}{2}+\frac{x}{55}}=50.\]](http://latex.artofproblemsolving.com/8/0/4/80419da878b6f43d2ce0eb55fdb332542584e9b5.png)
Simplifying the equation, we get ![\[15+x=25+\frac{10x}{11}.\]](http://latex.artofproblemsolving.com/7/7/c/77c48a48d74aa538bf0f66c7a122dfea6e814dba.png)
Solving the equation yields 
so our answer is 
.
Solution 3
If he travels miles at a speed of miles per hour, he travels for 30 min. Average rate is total distance over total time so 
, where d is the distance left to travel and t is the time to travel that distance. Solve for 
to get 
. You also know that he has to travel miles per hour for some time, so 
. Plug that in for d to get 
and 
and since , 
, the answer is .
Video Solution
https://www.youtube.com/watch?v=OC1KdFeZFeE
Associated Video
- happytwin
https://www.youtube.com/watch?v=0rcDe2bDRug
Video Solution
Solution detailing how to solve the problem:
https://www.youtube.com/watch?v=sEZ0sM-d1FA&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=17
Video Solution
- Soo, DRMS, NM
Video Solution
~savannahsolver
See Also
| 2019 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
