A baker uses $6\tfrac{2}{3}$ cups of flour when she prepares $\tfrac{5}{3}$ recipes of rolls. She will use $9\tfrac{3}{4}$ cups of flour when she prepares $\tfrac{m}{n}$ recipes of rolls where m and n are relatively prime positive integers. Find $m + n.$
1. Convert the mixed numbers to improper fractions:
   \[
   6 \frac{2}{3} = 6 + \frac{2}{3} = \frac{18}{3} + \frac{2}{3} = \frac{20}{3}
   \]
   \[
   9 \frac{3}{4} = 9 + \frac{3}{4} = \frac{36}{4} + \frac{3}{4} = \frac{39}{4}
   \]

2. Find the ratio between the two amounts of flour:
   \[
   \text{Ratio} = \frac{\frac{39}{4}}{\frac{20}{3}} = \frac{39}{4} \cdot \frac{3}{20} = \frac{39 \cdot 3}{4 \cdot 20} = \frac{117}{80}
   \]

3. Apply this ratio to the number of recipes:
   \[
   \frac{5}{3} \cdot \frac{117}{80} = \frac{5 \cdot 117}{3 \cdot 80} = \frac{585}{240}
   \]

4. Simplify the fraction \(\frac{585}{240}\):
   \[
   \frac{585}{240} = \frac{585 \div 15}{240 \div 15} = \frac{39}{16}
   \]

5. Identify \(m\) and \(n\) from the simplified fraction \(\frac{39}{16}\):
   \[
   m = 39, \quad n = 16
   \]

6. Add \(m\) and \(n\):
   \[
   m + n = 39 + 16 = 55
   \]

The final answer is \(\boxed{55}\).