1. The problem is to add the mixed numbers $3 \frac{1}{4}$ and $2 \frac{3}{5}$ and express the sum as a mixed number in simplest form. 2. Recall that to add mixed numbers, first convert them to improper fractions. 3. Convert $3 \frac{1}{4}$ to an improper fraction: $$3 \frac{1}{4} = \frac{3 \times 4 + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}$$ 4. Convert $2 \frac{3}{5}$ to an improper fraction: $$2 \frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5}$$ 5. Find a common denominator to add $\frac{13}{4}$ and $\frac{13}{5}$. The least common denominator (LCD) of 4 and 5 is 20. 6. Convert both fractions to have denominator 20: $$\frac{13}{4} = \frac{13 \times 5}{4 \times 5} = \frac{65}{20}$$ $$\frac{13}{5} = \frac{13 \times 4}{5 \times 4} = \frac{52}{20}$$ 7. Add the fractions: $$\frac{65}{20} + \frac{52}{20} = \frac{65 + 52}{20} = \frac{117}{20}$$ 8. Convert $\frac{117}{20}$ back to a mixed number by dividing 117 by 20: $$117 \div 20 = 5 \text{ remainder } 17$$ So, $$\frac{117}{20} = 5 \frac{17}{20}$$ 9. The fraction $\frac{17}{20}$ is already in simplest form because 17 and 20 have no common factors other than 1. 10. Therefore, the sum is: $$3 \frac{1}{4} + 2 \frac{3}{5} = 5 \frac{17}{20}$$