Alex launches his boat into a river and heads upstream at a constant speed. At the same time at a point $8$ miles upstream from Alex, Alice launches her boat and heads downstream at a constant speed. Both boats move at $6$ miles per hour in still water, but the river is  owing downstream at $2\frac{3}{10}$ miles per hour. Alex and Alice will meet at a point that is $\frac{m}{n}$ miles from Alex's starting point, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
1. **Determine the effective speeds of Alex and Alice:**
   - The river flows downstream at \(2\frac{3}{10} = \frac{23}{10}\) miles per hour.
   - Alex's speed in still water is \(6\) miles per hour. Since he is going upstream, his effective speed is:
     \[
     6 - \frac{23}{10} = 6 - 2.3 = 3.7 = \frac{37}{10} \text{ miles per hour}
     \]
   - Alice's speed in still water is \(6\) miles per hour. Since she is going downstream, her effective speed is:
     \[
     6 + \frac{23}{10} = 6 + 2.3 = 8.3 = \frac{83}{10} \text{ miles per hour}
     \]

2. **Set up the equation for the meeting point:**
   - Let \(d\) be the distance from Alex's starting point to the meeting point.
   - The distance Alice travels to the meeting point is \(8 - d\) miles.
   - The time it takes for both to meet is the same. Therefore, we can set up the equation:
     \[
     \frac{d}{\frac{37}{10}} = \frac{8 - d}{\frac{83}{10}}
     \]

3. **Solve the equation:**
   - Clear the fractions by multiplying both sides by \(10\):
     \[
     \frac{10d}{37} = \frac{10(8 - d)}{83}
     \]
   - Simplify:
     \[
     \frac{10d}{37} = \frac{80 - 10d}{83}
     \]
   - Cross-multiply to solve for \(d\):
     \[
     10d \cdot 83 = 37 \cdot (80 - 10d)
     \]
     \[
     830d = 2960 - 370d
     \]
     \[
     830d + 370d = 2960
     \]
     \[
     1200d = 2960
     \]
     \[
     d = \frac{2960}{1200} = \frac{37}{15}
     \]

4. **Express the distance in the form \(\frac{m}{n}\):**
   - The distance \(d\) from Alex's starting point to the meeting point is \(\frac{37}{15}\) miles.
   - Here, \(m = 37\) and \(n = 15\).

5. **Find \(m + n\):**
   \[
   m + n = 37 + 15 = 52
   \]

The final answer is \(\boxed{52}\).