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

Revision as of 11:03, 2 January 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(answer options)

The only option that is easily divisible by $55$ is $110$ . Which gives 2 hours of travel. And by the formula $\frac{15}{30} + \frac{110}{50} = \frac{5}{2}$

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

~phoenixfire

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

@@ Line 21: / Line 21: @@
 ==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 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\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

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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