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

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==Solution==
 
==Solution==
 
Let total distance be <math>x</math>. Her speed in miles per minute is <math>\tfrac{x}{180}</math>. Then, the distance that she drove before hitting the snowstorm is <math>\tfrac{x}{3}</math>. Her speed in snowstorm is reduced <math>20</math> miles per hour, or <math>\tfrac{1}{3}</math> miles per minute. Knowing it took her <math>276</math> minutes in total, we create equation:  
 
Let total distance be <math>x</math>. Her speed in miles per minute is <math>\tfrac{x}{180}</math>. Then, the distance that she drove before hitting the snowstorm is <math>\tfrac{x}{3}</math>. Her speed in snowstorm is reduced <math>20</math> miles per hour, or <math>\tfrac{1}{3}</math> miles per minute. Knowing it took her <math>276</math> minutes in total, we create equation:  
<cmath>\text{Time before Storm} + \text{Time after Storm} = \text{Total Time} \Longrightarrow \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</cmath>
+
<cmath>\text{Time before Storm}\, + \, \text{Time after Storm} = \text{Total Time} \Longrightarrow \\
 +
\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</cmath>
  
 
Solving equation, we get <math>x=135</math> <math>\Longrightarrow \boxed{B}</math>.
 
Solving equation, we get <math>x=135</math> <math>\Longrightarrow \boxed{B}</math>.

Revision as of 18:53, 8 February 2017

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 $\tfrac{x}{180}$. 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: \[\text{Time before Storm}\, + \, \text{Time after Storm} = \text{Total Time} \Longrightarrow \\  \frac{\text{Distance before Storm}}{\text{Speed before Storm}} + \frac{\text{Distance in Storm}}{\text{Speed in Storm}} = \text{Total Time} \Longrightarrow \frac{\tfrac{x}{3}}{\tfrac{x}{180}} + \frac{\tfrac{2x}{3}}{\tfrac{x}{180} - \tfrac{1}{3}} = 276\]

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

See Also

2017 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 12
Followed by
Problem 14
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 AMC 12 Problems and Solutions

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