2013 AMC 8 Problems/Problem 7

Problem

Trey and his mom stopped at a railroad crossing to let a train pass. As the train began to pass, Trey counted 6 cars in the first 10 seconds. It took the train 2 minutes and 45 seconds to clear the crossing at a constant speed. Which of the following was the most likely number of cars in the train?

$\textbf{(A)}\ 60 \qquad \textbf{(B)}\ 80 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 120 \qquad \textbf{(E)}\ 140$

Solution 1

Clearly, for every $5$ seconds, $3$ cars pass. It's more convenient to have everything in seconds: $2$ minutes and $45$ seconds$=2\cdot60 + 45 = 165$ seconds. We then set up a ratio: $$\frac{3}{5}=\frac{x}{165}$$ $$3(165)=5x$$ $$x=3(33)=99\approx\boxed{\textbf{(C)}\ 100}.$$

~megaboy6679 ~MiracleMaths

Solution 2

$2$ minutes and $45$ seconds is equal to $120+45=165\text{ seconds}$.

Since $6$ cars pass at around $10$ seconds, there are about $\left \lfloor{\dfrac{165}{10}}\right \rfloor =16$ groups of $6$ cars. There are about $16\cdot6=96\text{ cars}$, so the closest answer choice is $\boxed{\textbf{(C)}\ 100}$.

Video Solution

https://youtu.be/79lJItxCt50 ~savannahsolver