A train traveling at $ 80$ mph begins to cross a $ 1$ mile long bridge. At this moment, a man begins to walk from the front of the train to the back of the train at a speed of $5$ mph. The man reaches the back of the train as soon as the train is completely off the bridge. What is the length of the train (as a fraction of a mile)?
1. **Determine the time it takes for the train to cross the bridge:**
   - The train is traveling at a speed of \(80\) mph.
   - The bridge is \(1\) mile long.
   - Time taken to cross the bridge is given by:
     \[
     \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{1 \text{ mile}}{80 \text{ mph}} = \frac{1}{80} \text{ hours}
     \]

2. **Determine the distance the man walks:**
   - The man starts walking from the front to the back of the train at a speed of \(5\) mph.
   - The man walks for the same amount of time it takes the train to cross the bridge, which is \(\frac{1}{80}\) hours.
   - The distance the man walks is given by:
     \[
     \text{Distance} = \text{Speed} \times \text{Time} = 5 \text{ mph} \times \frac{1}{80} \text{ hours} = \frac{5}{80} \text{ miles} = \frac{1}{16} \text{ miles}
     \]

3. **Interpret the result:**
   - The man walks from the front to the back of the train, covering the length of the train.
   - Therefore, the length of the train is \(\frac{1}{16}\) miles.

The final answer is \(\boxed{\frac{1}{16} \text{ miles}}\).