Difference between revisions of "2019 AMC 8 Problems/Problem 16"

(Video Solution)
m (Solution 2)
(8 intermediate revisions by 7 users not shown)
Line 14: Line 14:
  
 
==Solution 2==
 
==Solution 2==
Note that the average speed is simply the total distance over the total time. Let the number of additional miles he has to drive be <math>x.</math> Therefore, the total distance is <math>15+x</math> and the total time (in hours) is <cmath>\frac{15}{30}+\frac{x}{55}=\frac{1}{2}+\frac{x}{55}.</cmath> We can set up the following equation: <cmath>\frac{15+x}{\frac{1}{2}+\frac{x}{55}}=50.</cmath> Simplifying the equation, we get <cmath>15+x=25+\frac{10x}{11}.</cmath> Solving the equation yields <math>x=110,</math> so our answer is <math>\boxed{\textbf{(D)}\ 110}</math>.  
+
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 <math>x.</math> Therefore, the total distance is <math>15+x</math> and the total time (in hours) is <cmath>\frac{15}{30}+\frac{x}{55}=\frac{1}{2}+\frac{x}{55}.</cmath> We can set up the following equation: <cmath>\frac{15+x}{\frac{1}{2}+\frac{x}{55}}=50.</cmath> Simplifying the equation, we get <cmath>15+x=25+\frac{10x}{11}.</cmath> Solving the equation yields <math>x=110,</math> so our answer is <math>\boxed{\textbf{(D)}\ 110}</math>.
 
 
~twinemma
 
  
 
==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>.
  
-goldenn
+
==Video Solution==
 +
 
 +
Associated Video - https://www.youtube.com/watch?v=OC1KdFeZFeE
 +
 
 +
https://youtu.be/5K1AgeZ8rUQ - 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==
 
==Video Solution==
 +
https://youtu.be/aFsC5awOWBk
 +
Soo, DRMS, NM
  
https://youtu.be/5K1AgeZ8rUQ - happytwin
+
==Video Solution==
 +
https://youtu.be/btmFN_C1zSg
  
Video Solution - https://youtu.be/Lw8fSbX_8FU (Also explains problems 11-20)
+
~savannahsolver
  
 
==See Also==
 
==See Also==

Revision as of 15:31, 4 August 2022

Problem 16

Qiang drives $15$ miles at an average speed of $30$ miles per hour. How many additional miles will he have to drive at $55$ miles per hour to average $50$ miles per hour for the entire trip?

$\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135$

Solution 1

The only option that is easily divisible by $55$ is $110$. Which gives 2 hours of travel. And by the formula $\frac{15}{30} + \frac{110}{55} = \frac{5}{2}$

And $\text{Average Speed}$ = $\frac{\text{Total Distance}}{\text{Total Time}}$

Thus $\frac{125}{50} = \frac{5}{2}$

Both are equal and thus our answer is $\boxed{\textbf{(D)}\ 110}.$

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 $x.$ Therefore, the total distance is $15+x$ and the total time (in hours) is \[\frac{15}{30}+\frac{x}{55}=\frac{1}{2}+\frac{x}{55}.\] We can set up the following equation: \[\frac{15+x}{\frac{1}{2}+\frac{x}{55}}=50.\] Simplifying the equation, we get \[15+x=25+\frac{10x}{11}.\] Solving the equation yields $x=110,$ so our answer is $\boxed{\textbf{(D)}\ 110}$.

Solution 3

If he travels $15$ miles at a speed of $30$ miles per hour, he travels for 30 min. Average rate is total distance over total time so $(15+d)/(0.5 + t) = 50$, where d is the distance left to travel and t is the time to travel that distance. solve for $d$ to get $d = 10+50t$. you also know that he has to travel $55$ miles per hour for some time, so $d=55t$ plug that in for d to get $55t = 10+50t$ and $t=2$ and since $d=55t$, $d = 2\cdot55 =110$ the answer is $\boxed{\textbf{(D)}\ 110}$.

Video Solution

Associated Video - https://www.youtube.com/watch?v=OC1KdFeZFeE

https://youtu.be/5K1AgeZ8rUQ - 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

https://youtu.be/aFsC5awOWBk Soo, DRMS, NM

Video Solution

https://youtu.be/btmFN_C1zSg

~savannahsolver

See Also

2019 AMC 8 (ProblemsAnswer KeyResources)
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. AMC logo.png