1. The problem is to solve the expression involving mixed numbers: $1 \frac{3}{10} + 2 \frac{7}{5} + 3 \frac{2}{3}$.\n\n2. Convert each mixed number to an improper fraction:\n- $1 \frac{3}{10} = \frac{10}{10} + \frac{3}{10} = \frac{13}{10}$\n- $2 \frac{7}{5} = 2 + \frac{7}{5} = \frac{10}{5} + \frac{7}{5} = \frac{17}{5}$\n- $3 \frac{2}{3} = 3 + \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3}$\n\n3. Find the least common denominator (LCD) for the fractions $\frac{13}{10}$, $\frac{17}{5}$, and $\frac{11}{3}$.\n- The denominators are 10, 5, and 3.\n- The LCD of 10, 5, and 3 is 30.\n\n4. Convert each fraction to have denominator 30:\n- $\frac{13}{10} = \frac{13 \times 3}{10 \times 3} = \frac{39}{30}$\n- $\frac{17}{5} = \frac{17 \times 6}{5 \times 6} = \frac{102}{30}$\n- $\frac{11}{3} = \frac{11 \times 10}{3 \times 10} = \frac{110}{30}$\n\n5. Add the fractions:\n$$\frac{39}{30} + \frac{102}{30} + \frac{110}{30} = \frac{39 + 102 + 110}{30} = \frac{251}{30}$$\n\n6. Convert the improper fraction back to a mixed number:\n- Divide 251 by 30: $251 \div 30 = 8$ remainder $11$.\n- So, $\frac{251}{30} = 8 \frac{11}{30}$.\n\n7. Final answer: $8 \frac{11}{30}$.