1. The problem is to find the sum of the mixed numbers $6 \frac{3}{4}$ and $6 \frac{3}{4}$. 2. First, convert the mixed numbers to improper fractions. $$6 \frac{3}{4} = 6 + \frac{3}{4} = \frac{6 \times 4}{4} + \frac{3}{4} = \frac{24}{4} + \frac{3}{4} = \frac{27}{4}$$ 3. Now add the two improper fractions: $$\frac{27}{4} + \frac{27}{4} = \frac{27 + 27}{4} = \frac{54}{4}$$ 4. Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2: $$\frac{\cancel{54}^{{27}}}{\cancel{4}^{{2}}} = \frac{27}{2}$$ 5. Convert the improper fraction back to a mixed number: $$\frac{27}{2} = 13 \frac{1}{2}$$ 6. Therefore, the sum of $6 \frac{3}{4}$ and $6 \frac{3}{4}$ is $13 \frac{1}{2}$.