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

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==Problem 16==
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==Problem==
 
Qiang drives <math>15</math> miles at an average speed of <math>30</math> miles per hour. How many additional miles will he have to drive at <math>55</math> miles per hour to average <math>50</math> miles per hour for the entire trip?
 
Qiang drives <math>15</math> miles at an average speed of <math>30</math> miles per hour. How many additional miles will he have to drive at <math>55</math> miles per hour to average <math>50</math> miles per hour for the entire trip?
  
 
<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>
  
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==Solution 1(answer options)==
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==Solution 1 (by looking at the answer choices)==
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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>
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The only option that is easily divisible by <math>55</math> is <math>110</math>, which gives 2 hours of travel. And, the formula is <math>\frac{15}{30} + \frac{110}{55} = \frac{5}{2}</math>.
  
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And <math>\text{Average Speed}</math> = <math>\frac{\text{Total Distance}}{\text{Total Time}}</math>
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And, <math>\text{Average Speed}</math> = <math>\frac{\text{Total Distance}}{\text{Total Time}}</math>.
 
   
 
   
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Thus <math>\frac{125}{50} = \frac{5}{2}</math>
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Thus, <math>\frac{125}{50} = \frac{5}{2}</math>.
  
 
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>
  
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~phoenixfire
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==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 <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>.
  
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==Solution 2==
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==Solution 3==
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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>.
+
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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 +
==Solution 4==  
 +
 
 +
Let <math>h</math> be the amount of hours Qiang drives after his first 15 miles. Average speed, which we know is <math>50</math> mph, means total distance over total time. For 15 miles at 30 mph, the time taken is <math>\frac{1}{2}</math> hour, so the total time for this trip would be <math>\frac{1}{2} + h</math> hours. For the total distance, 15 miles are traveled in the first part and <math>55h</math> miles in the second. This gives the following equation:  
 +
 
 +
 
 +
<cmath>\dfrac{15+55h}{\frac{1}{2}+h} = 50.</cmath>  
 +
 
 +
 
 +
 
 +
 
 +
Cross multiplying, we get that <math>15 + 55h = 50h + 25</math>, and simple algebra gives <math>h=2</math>. In 2 hours traveling at 55 mph, the distance traveled is <math>\frac{2 \hspace{0.05 in} \text{hours}}{1} \cdot \frac{55 \hspace{0.05 in} \text{miles}}{1 \hspace{0.05 in} \text{hour}} = 2 \cdot 55 \hspace{0.05 in} \text{miles} = 110 \hspace{0.05 in} \text{miles}</math>, which is choice <math>\boxed{\textbf{(D)}\ 110}</math>.  
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 +
 
 +
 
 +
 
 +
~TaeKim
 +
 
 +
==Video Solution==
 +
 
 +
==Video Solution by Math-X (First fully understand the problem!!!)==
 +
https://youtu.be/IgpayYB48C4?si=8YldGqXbPzZfeA-z&t=4756
 +
 
 +
~Math-X
 +
 
 +
 
 +
https://www.youtube.com/watch?v=OC1KdFeZFeE
 +
 
 +
Associated Video
 +
 
 +
https://youtu.be/5K1AgeZ8rUQ
 +
 
 +
- happytwin
 +
 
 +
https://www.youtube.com/watch?v=0rcDe2bDRug  ~David
 +
 
 +
== 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
 +
 
 +
==Video Solution (The Fastest Way)==
 +
https://www.youtube.com/watch?v=IOhkO3c3c2A&ab_channel=SaxStreak002
  
−
~twinemma
+
~SaxStreak
  
 +
==Video Solution (MOST EFFICIENT+ CREATIVE THINKING!!!(BTW WE LOVE THIS ONE))==
 +
https://youtu.be/JiTos1fFtUA
  
 +
~Education, the Study of Everything
  
−
==Solution 3==
+
==Video Solution by The Power of Logic(1 to 25 Full Solution)==
−
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 tragvel that distance. solve for <math>d</math> to get <math>d = 10+50t</math>. you also know that he has to tral <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*55 =110</math> the answer is <math>\boxed{\textbf{(D)}\ 110}</math>.
+
https://youtu.be/Xm4ZGND9WoY
  
−
-goldenn
+
~Hayabusa1
  
 
==See Also==
 
==See Also==

Latest revision as of 09:32, 9 November 2024

Problem

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 (by looking at the answer choices)

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

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

Solution 4

Let $h$ be the amount of hours Qiang drives after his first 15 miles. Average speed, which we know is $50$ mph, means total distance over total time. For 15 miles at 30 mph, the time taken is $\frac{1}{2}$ hour, so the total time for this trip would be $\frac{1}{2} + h$ hours. For the total distance, 15 miles are traveled in the first part and $55h$ miles in the second. This gives the following equation:


\[\dfrac{15+55h}{\frac{1}{2}+h} = 50.\]



Cross multiplying, we get that $15 + 55h = 50h + 25$, and simple algebra gives $h=2$. In 2 hours traveling at 55 mph, the distance traveled is $\frac{2 \hspace{0.05 in} \text{hours}}{1} \cdot \frac{55 \hspace{0.05 in} \text{miles}}{1 \hspace{0.05 in} \text{hour}} = 2 \cdot 55 \hspace{0.05 in} \text{miles} = 110 \hspace{0.05 in} \text{miles}$, which is choice $\boxed{\textbf{(D)}\ 110}$.



~TaeKim

Video Solution

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/IgpayYB48C4?si=8YldGqXbPzZfeA-z&t=4756

~Math-X


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

Associated Video

https://youtu.be/5K1AgeZ8rUQ

- happytwin

https://www.youtube.com/watch?v=0rcDe2bDRug ~David

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

Video Solution (The Fastest Way)

https://www.youtube.com/watch?v=IOhkO3c3c2A&ab_channel=SaxStreak002

~SaxStreak

Video Solution (MOST EFFICIENT+ CREATIVE THINKING!!!(BTW WE LOVE THIS ONE))

https://youtu.be/JiTos1fFtUA

~Education, the Study of Everything

Video Solution by The Power of Logic(1 to 25 Full Solution)

https://youtu.be/Xm4ZGND9WoY

~Hayabusa1

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

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