1. The problem is to add the mixed number $1 \frac{2}{3}$ and the fraction $\frac{3}{5}$. 2. First, convert the mixed number to an improper fraction. The formula is $a \frac{b}{c} = \frac{ac + b}{c}$. Here, $a=1$, $b=2$, and $c=3$. 3. Calculate the improper fraction: $$1 \frac{2}{3} = \frac{1 \times 3 + 2}{3} = \frac{3 + 2}{3} = \frac{5}{3}.$$ 4. Now add $\frac{5}{3}$ and $\frac{3}{5}$. To add fractions, find a common denominator. The denominators are 3 and 5, so the least common denominator (LCD) is 15. 5. Convert each fraction to have denominator 15: $$\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}, \quad \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}.$$ 6. Add the fractions: $$\frac{25}{15} + \frac{9}{15} = \frac{25 + 9}{15} = \frac{34}{15}.$$ 7. Convert the improper fraction back to a mixed number by dividing numerator by denominator: $$34 \div 15 = 2 \text{ remainder } 4,$$ so $$\frac{34}{15} = 2 \frac{4}{15}.$$ 8. Final answer: $$1 \frac{2}{3} + \frac{3}{5} = 2 \frac{4}{15}.$$