2019 AMC 8

Complete problem set with solutions and individual problem pages

Problem 16 Hard

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?

  • A.

    45

  • B.

    62

  • C.

    90

  • D.

    110

  • E.

    135

Answer:D

Solution 1

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}.

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