A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.
1. **Identify the problem**: We need to find the probability that a point randomly selected from inside a rectangle with side lengths 6 and 8 is closer to a side of the rectangle than to either diagonal.

2. **Calculate the area of the rectangle**: 
   \[
   \text{Area of the rectangle} = 6 \times 8 = 48
   \]

3. **Determine the regions closer to the sides than to the diagonals**: 
   - The diagonals of the rectangle divide it into four right triangles.
   - We need to find the inradii of these triangles to determine the regions closer to the sides.

4. **Calculate the inradii of the triangles**:
   - Each triangle has one side equal to 6, one side equal to 8, and the hypotenuse equal to 10 (since \( \sqrt{6^2 + 8^2} = 10 \)).
   - The area of each triangle is:
     \[
     \text{Area} = \frac{1}{2} \times 6 \times 8 = 24
     \]
   - The semiperimeter \( s \) of each triangle is:
     \[
     s = \frac{6 + 8 + 10}{2} = 12
     \]
   - The inradius \( r \) of each triangle is given by:
     \[
     r = \frac{\text{Area}}{s} = \frac{24}{12} = 2
     \]

5. **Calculate the total area closer to the sides**:
   - The total area closer to the sides is the area of the rectangle minus the area closer to the diagonals.
   - The area closer to the diagonals is the area of the four triangles with inradius 2.
   - Each triangle has an area of 24, so the total area of the four triangles is:
     \[
     4 \times 24 = 96
     \]
   - However, this is incorrect because the triangles overlap. We need to consider the correct regions.

6. **Correct calculation of the regions**:
   - The correct approach is to consider the regions within the rectangle that are closer to the sides.
   - The regions closer to the sides form a smaller rectangle within the original rectangle.
   - The dimensions of this smaller rectangle are \( 6 - 2 \times 2 \) and \( 8 - 2 \times 2 \):
     \[
     (6 - 4) \times (8 - 4) = 2 \times 4 = 8
     \]

7. **Calculate the probability**:
   - The area closer to the sides is:
     \[
     48 - 8 = 40
     \]
   - The probability is:
     \[
     \frac{40}{48} = \frac{5}{6}
     \]

8. **Express the probability in terms of relatively prime integers \( m \) and \( n \)**:
   - Here, \( m = 5 \) and \( n = 6 \).

9. **Find \( m + n \)**:
   \[
   m + n = 5 + 6 = 11
   \]

The final answer is \( \boxed{11} \)