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?
To solve this problem, we need to determine the speed during the return trip such that the average speed for the entire round trip is an integer. Let's denote the speed during the return trip as \( v \) miles per hour.

1. **Define the variables and the given conditions:**
   - Speed to the destination: \( 50 \) miles per hour.
   - Speed on the return trip: \( v \) miles per hour.
   - The average speed for the round trip is an integer.

2. **Calculate the total distance and total time:**
   - Let the distance to the destination be \( d \) miles.
   - The total distance for the round trip is \( 2d \) miles.
   - Time to the destination: \( \frac{d}{50} \) hours.
   - Time for the return trip: \( \frac{d}{v} \) hours.

3. **Calculate the average speed for the round trip:**
   The average speed \( \bar{v} \) is given by the total distance divided by the total time:
   \[
   \bar{v} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{50} + \frac{d}{v}}
   \]
   Simplify the expression:
   \[
   \bar{v} = \frac{2d}{\frac{d(50 + v)}{50v}} = \frac{2d \cdot 50v}{d(50 + v)} = \frac{100v}{50 + v}
   \]

4. **Ensure the average speed is an integer:**
   For \( \bar{v} \) to be an integer, \( \frac{100v}{50 + v} \) must be an integer. Let \( \bar{v} = k \), where \( k \) is an integer:
   \[
   k = \frac{100v}{50 + v}
   \]
   Rearrange to solve for \( v \):
   \[
   k(50 + v) = 100v \implies 50k + kv = 100v \implies 50k = v(100 - k)
   \]
   \[
   v = \frac{50k}{100 - k}
   \]

5. **Find integer solutions for \( v \) within the given constraints:**
   Since \( v \) must be an integer and \( v \leq 100 \):
   \[
   \frac{50k}{100 - k} \leq 100
   \]
   Simplify:
   \[
   50k \leq 100(100 - k) \implies 50k \leq 10000 - 100k \implies 150k \leq 10000 \implies k \leq \frac{10000}{150} \implies k \leq \frac{200}{3} \approx 66.67
   \]
   Therefore, \( k \) must be an integer less than or equal to 66.

6. **Check for valid \( k \) values:**
   We need \( \frac{50k}{100 - k} \) to be an integer. Let's test some values of \( k \):
   - For \( k = 25 \):
     \[
     v = \frac{50 \cdot 25}{100 - 25} = \frac{1250}{75} = \frac{50}{3} \quad \text{(not an integer)}
     \]
   - For \( k = 40 \):
     \[
     v = \frac{50 \cdot 40}{100 - 40} = \frac{2000}{60} = \frac{100}{3} \quad \text{(not an integer)}
     \]
   - For \( k = 60 \):
     \[
     v = \frac{50 \cdot 60}{100 - 60} = \frac{3000}{40} = 75 \quad \text{(integer and valid)}
     \]

Thus, the speed during the return trip must be \( 75 \) miles per hour.

The final answer is \( \boxed{75} \).