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

(Solution 2)
(Video Solution)
(7 intermediate revisions by 6 users not shown)
Line 4: Line 4:
 
<math>\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135</math>
 
<math>\textbf{(A) }45\qquad\textbf{(B) }62\qquad\textbf{(C) }90\qquad\textbf{(D) }110\qquad\textbf{(E) }135</math>
  
==Solution 1(answer options)==
+
==Solution 1==
The only option that is easily divisible by <math>55</math> is <math>110</math>. Which gives 2 hours of travel. And by the formula <math>\frac{15}{30} + \frac{110}{50} = \frac{5}{2}</math>
+
The only option that is easily divisible by <math>55</math> is <math>110</math>. Which gives 2 hours of travel. And by the formula <math>\frac{15}{30} + \frac{110}{55} = \frac{5}{2}</math>
  
 
And <math>\text{Average Speed}</math> = <math>\frac{\text{Total Distance}}{\text{Total Time}}</math>
 
And <math>\text{Average Speed}</math> = <math>\frac{\text{Total Distance}}{\text{Total Time}}</math>
Line 12: Line 12:
  
 
Both are equal and thus our answer is <math>\boxed{\textbf{(D)}\ 110}.</math>
 
Both are equal and thus our answer is <math>\boxed{\textbf{(D)}\ 110}.</math>
 
~phoenixfire
 
  
 
==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>.  
 
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>.  
  
~TwinEmma
+
~twinemma
  
 
==Solution 3==
 
==Solution 3==
Line 24: Line 22:
  
 
-goldenn
 
-goldenn
 +
 +
==Video Solution==
 +
 +
Associated Video - https://www.youtube.com/watch?v=OC1KdFeZFeE
 +
 +
https://youtu.be/5K1AgeZ8rUQ - happytwin
 +
 +
Video Solution - https://youtu.be/Lw8fSbX_8FU (Also explains problems 11-20)
  
 
==See Also==
 
==See Also==

Revision as of 12:08, 1 October 2020

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

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 $x.$ 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 $\boxed{\textbf{(D)}\ 110}$.

~twinemma

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 $\boxed{\textbf{(D)}\ 110}$.

-goldenn

Video Solution

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

https://youtu.be/5K1AgeZ8rUQ - happytwin

Video Solution - https://youtu.be/Lw8fSbX_8FU (Also explains problems 11-20)

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

Invalid username
Login to AoPS