1. The problem is to add the mixed numbers $3 \frac{3}{10}$ and $2 \frac{1}{5}$. 2. First, convert the mixed numbers to improper fractions. $$3 \frac{3}{10} = \frac{3 \times 10 + 3}{10} = \frac{30 + 3}{10} = \frac{33}{10}$$ $$2 \frac{1}{5} = \frac{2 \times 5 + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5}$$ 3. To add these fractions, find a common denominator. The denominators are 10 and 5. The least common denominator (LCD) is 10. 4. Convert $\frac{11}{5}$ to have denominator 10: $$\frac{11}{5} = \frac{11 \times 2}{5 \times 2} = \frac{22}{10}$$ 5. Now add the fractions: $$\frac{33}{10} + \frac{22}{10} = \frac{33 + 22}{10} = \frac{55}{10}$$ 6. Simplify the fraction $\frac{55}{10}$ by dividing numerator and denominator by 5: $$\frac{\cancel{55}^{11}}{\cancel{10}^2} = \frac{11}{2}$$ 7. Convert the improper fraction back to a mixed number: $$\frac{11}{2} = 5 \frac{1}{2}$$ 8. Therefore, the sum of $3 \frac{3}{10}$ and $2 \frac{1}{5}$ is $5 \frac{1}{2}$.