# 2008 iTest Problems/Problem 33

## Problem

One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel $50$ miles per hour while heading to our final vacation destination..." Hannah teases her husband, "You $\textit{would}$ drive that $\textit{slowly}\text{!}$"

Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we travel at a constant rate of $50$ miles per hour on the way to the beach. However, on the way back we travel at a faster constant rate along the exact same route. If our faster return rate is an integer number of miles per hour, and our average speed for the $\textit{whole round trip}$ is $\textit{also}$ an integer number of miles per hour, what must be our speed during the return trip?" Michael pipes up, "How about $4950$ miles per hour?!" Wendy smiles, "For the sake of your $\textit{other}$ children, please don't let $\textit{Michael}$ drive." Jerry adds, "How about we assume that we never $\textit{ever}$ drive more than $100$ miles per hour. Michael and Wendy, let Josh and Alexis try this one." Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate. But Alexis scribbles some work on her napkin and declares the correct answer. What answer did Alexis find?

## Solution

Let $t$ be the time (in hours) it takes to drive on the way to the final vacation destination. That means the distance to the destination is $50t$ miles, and the time the family takes on the way back is $\tfrac{50t}{r}$ hours, where $r$ is the return trip rate in mph. Also, let the average rate of the trip be $n$ mph, where $n$ is an integer. Using the distance-rate-time formula, $$\frac{50t+50t}{t+\frac{50t}{r}} = n$$ $$\frac{100}{1+\frac{50}{r}} = n$$ $$100 = n + \frac{50n}{r}$$ $$100r = nr + 50n$$ $$0 = nr - 100r + 50n$$ Using Simon's Favorite Factoring Trick, $$0 = (n-100)(r+50) + 5000$$ $$5000 = (100-n)(r+50)$$ Going through each factor of $5000$ and solving for $n$ and $r$, the only value $r$ less than $100$ that makes $n$ an integer is $75$. The speed of the return trip is $\boxed{75}$ mph.