2019 AMC 8 Problems/Problem 16

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 $\text{Average Speed}$ = $\frac{\text{Total Distance}{{\text{Total Time}}$ (Error compiling LaTeX. Unknown error_msg)

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

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 15	Followed by Problem 17
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2019 AMC 8 Problems/Problem 16

Contents

Problem 16

Solution 1(answer options)

Solution 2

See Also