# Difference between revisions of "2018 AMC 10B Problems/Problem 2"

## Problem

Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?

$\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$

## Solution 1

Let Sam drive at exactly $60$ mph in the first half hour, $65$ mph in the second half hour, and $x$ mph in the third half hour.

Due to $rt = d$, and that $30$ min is half an hour, he covered $60 \cdot \frac{1}{2} = 30$ miles in the first $30$ mins.

SImilarly, he covered $\frac{65}{2}$ miles in the $2$nd half hour period.

The problem states that Sam drove $96$ miles in $90$ min, so that means that he must have covered $96 - \left(30 + \frac{65}{2}\right) = 33 \frac{1}{2}$ miles in the third half hour period.

$rt = d$, so $x \cdot \frac{1}{2} = 33 \frac{1}{2}$.

Therefore, Sam was driving $\boxed{\textbf{(D) } 67}$ miles per hour in the third half hour.

## Solution 2 (Faster)

The average speed for the total trip is $$\text{avg. speed} = \frac{96}{\frac{3}{2}} = 64.$$ Therefore the average speed for the total trip is the average of the average speeds of the three intrevals. So we have $64 = \frac{60 + 65 + x}{3}$ and solving for $x = 67$. So the answer is $\boxed{\textbf{(D) } 67}$.

~savannahsolver