1. **State the problem:** Add the fractions and mixed numbers $\frac{21}{4} + 3 \frac{3}{4}$. 2. **Convert mixed number to improper fraction:** $3 \frac{3}{4} = \frac{3 \times 4 + 3}{4} = \frac{12 + 3}{4} = \frac{15}{4}$. 3. **Add the fractions:** $\frac{21}{4} + \frac{15}{4} = \frac{21 + 15}{4} = \frac{36}{4}$. 4. **Simplify the fraction:** $\frac{36}{4} = \frac{\cancel{36}^{9}}{\cancel{4}^{1}} = 9$. 5. **Final answer:** $$9$$