Mary mixes $2$ gallons of a solution that is $40$ percent alcohol with $3$ gallons of a solution that is $60$ percent alcohol. Sandra mixes $4$ gallons of a solution that is $30$ percent alcohol with $\frac{m}{n}$ gallons of a solution that is $80$ percent alcohol, where $m$ and $n$ are relatively prime positive integers. Mary and Sandra end up with solutions that are the same percent alcohol. Find $m + n$.
1. **Calculate the total amount of alcohol in Mary's solution:**
   - Mary mixes 2 gallons of a 40% alcohol solution and 3 gallons of a 60% alcohol solution.
   - The amount of alcohol in the 2 gallons of 40% solution is:
     \[
     2 \times 0.40 = 0.8 \text{ gallons}
     \]
   - The amount of alcohol in the 3 gallons of 60% solution is:
     \[
     3 \times 0.60 = 1.8 \text{ gallons}
     \]
   - The total amount of alcohol in Mary's solution is:
     \[
     0.8 + 1.8 = 2.6 \text{ gallons}
     \]
   - The total volume of Mary's solution is:
     \[
     2 + 3 = 5 \text{ gallons}
     \]
   - The percentage of alcohol in Mary's solution is:
     \[
     \frac{2.6}{5} \times 100\% = 52\%
     \]

2. **Set up the equation for Sandra's solution:**
   - Sandra mixes 4 gallons of a 30% alcohol solution with \(\frac{m}{n}\) gallons of an 80% alcohol solution.
   - The amount of alcohol in the 4 gallons of 30% solution is:
     \[
     4 \times 0.30 = 1.2 \text{ gallons}
     \]
   - The amount of alcohol in the \(\frac{m}{n}\) gallons of 80% solution is:
     \[
     \frac{m}{n} \times 0.80 = \frac{0.8m}{n} \text{ gallons}
     \]
   - The total amount of alcohol in Sandra's solution is:
     \[
     1.2 + \frac{0.8m}{n} \text{ gallons}
     \]
   - The total volume of Sandra's solution is:
     \[
     4 + \frac{m}{n} \text{ gallons}
     \]

3. **Set up the equation for the percentage of alcohol in Sandra's solution:**
   - The percentage of alcohol in Sandra's solution must be equal to the percentage of alcohol in Mary's solution, which is 52%.
   - Therefore, we set up the equation:
     \[
     \frac{1.2 + \frac{0.8m}{n}}{4 + \frac{m}{n}} = 0.52
     \]

4. **Solve the equation:**
   - Multiply both sides by \(4 + \frac{m}{n}\) to clear the denominator:
     \[
     1.2 + \frac{0.8m}{n} = 0.52 \left(4 + \frac{m}{n}\right)
     \]
   - Distribute the 0.52 on the right-hand side:
     \[
     1.2 + \frac{0.8m}{n} = 0.52 \times 4 + 0.52 \times \frac{m}{n}
     \]
     \[
     1.2 + \frac{0.8m}{n} = 2.08 + 0.52 \times \frac{m}{n}
     \]
   - Subtract \(0.52 \times \frac{m}{n}\) from both sides:
     \[
     1.2 + \frac{0.8m}{n} - 0.52 \times \frac{m}{n} = 2.08
     \]
     \[
     1.2 + \left(\frac{0.8m}{n} - \frac{0.52m}{n}\right) = 2.08
     \]
     \[
     1.2 + \frac{0.28m}{n} = 2.08
     \]
   - Subtract 1.2 from both sides:
     \[
     \frac{0.28m}{n} = 0.88
     \]
   - Multiply both sides by \(\frac{n}{0.28}\):
     \[
     m = \frac{0.88n}{0.28}
     \]
     \[
     m = \frac{88n}{28}
     \]
     \[
     m = \frac{44n}{14}
     \]
     \[
     m = \frac{22n}{7}
     \]
   - Since \(m\) and \(n\) are relatively prime, we have \(m = 22\) and \(n = 7\).

5. **Find \(m + n\):**
   \[
   m + n = 22 + 7 = 29
   \]

The final answer is \(\boxed{29}\).