Difference between revisions of "2017 AMC 12A Problems/Problem 13"

Problem

Driving at a constant speed, Sharon usually takes $180$ minutes to drive from her house to her mother's house. One day Sharon begins the drive at her usual speed, but after driving $\frac{1}{3}$ of the way, she hits a bad snowstorm and reduces her speed by $20$ miles per hour. This time the trip takes her a total of $276$ minutes. How many miles is the drive from Sharon's house to her mother's house?

$\textbf{(A)}\ 132 \qquad\textbf{(B)}\ 135 \qquad\textbf{(C)}\ 138 \qquad\textbf{(D)}\ 141 \qquad\textbf{(E)}\ 144$

Solution

Let total distance be $x$. Her speed in miles per minute is $x/180$. Then, the distance that she drove before hitting the snowstorm is \tfrac{x}{3}. Her speed in snowstorm is reduced 20 miles per hour, or \tfrac{1}{3} miles per minute. Knowing it took her 276 minutes in total, we create equation: $$\frac{\text{Distance before Storm}}{\text{Speed before Storm}} + \frac{\text{Distance in Storm}}{\text{Speed in Storm}} = \text{Total Time} \Longrightarrow \frac{\tfrac{x}{3}}{\tfrac{x}{180}} + \frac{\tfrac{2x}{3}}{\tfrac{x}{180} - \tfrac{1}{3}} = 276$$

Solving equation, we get $x=135$ $\Longrightarrow \boxed{B}$.