 Given that real numbers $a$, $b$, and $c$ satisfy $ab=3$, $ac=4$, and $b+c=5$, the value of $bc$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.


[i]2021 CCA Math Bonanza Team Round #2[/i]
1. We start with the given equations:
   \[
   ab = 3, \quad ac = 4, \quad b + c = 5
   \]

2. We add and factor the equations involving \(a\):
   \[
   a(b + c) = ab + ac = 3 + 4 = 7
   \]
   Since \(b + c = 5\), we substitute this into the equation:
   \[
   a \cdot 5 = 7 \implies a = \frac{7}{5}
   \]

3. Next, we use the value of \(a\) to find \(b\) and \(c\). From \(ab = 3\):
   \[
   \left(\frac{7}{5}\right)b = 3 \implies b = \frac{3 \cdot 5}{7} = \frac{15}{7}
   \]

4. Using \(b + c = 5\) to find \(c\):
   \[
   \frac{15}{7} + c = 5 \implies c = 5 - \frac{15}{7} = \frac{35}{7} - \frac{15}{7} = \frac{20}{7}
   \]

5. Now, we calculate \(bc\):
   \[
   bc = \left(\frac{15}{7}\right) \left(\frac{20}{7}\right) = \frac{15 \cdot 20}{7 \cdot 7} = \frac{300}{49}
   \]

6. The fraction \(\frac{300}{49}\) is already in its simplest form since 300 and 49 have no common factors other than 1. Therefore, \(m = 300\) and \(n = 49\).

7. Finally, we compute \(m + n\):
   \[
   m + n = 300 + 49 = 349
   \]

The final answer is \(\boxed{349}\).