1. The problem asks us to find the value of $1 + \frac{2}{3} - \frac{4}{5}$.\n2. To add and subtract these fractions, we need a common denominator. The denominators are 1, 3, and 5. The least common denominator (LCD) is 15.\n3. Convert each term to have denominator 15:\n\n$1 = \frac{15}{15}$\n$\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}$\n$\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}$\n\n4. Substitute the equivalent fractions into the expression:\n\n$\frac{15}{15} + \frac{10}{15} - \frac{12}{15} = \frac{15 + 10 - 12}{15} = \frac{13}{15}$\n\n5. The result $\frac{13}{15}$ is in simplest form, so the final answer is $\frac{13}{15}$.