1. The problem is to add the fractions $\frac{7}{8}$ and $\frac{1}{3}$.\n2. To add fractions, first find a common denominator. The denominators are 8 and 3. The least common multiple (LCM) of 8 and 3 is 24.\n3. Convert each fraction to an equivalent fraction with denominator 24:\n\n$$\frac{7}{8} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}$$\n$$\frac{1}{3} = \frac{1 \times 8}{3 \times 8} = \frac{8}{24}$$\n\n4. Now, add the fractions with the same denominator:\n\n$$\frac{21}{24} + \frac{8}{24} = \frac{21 + 8}{24} = \frac{29}{24}$$\n\n5. The sum is an improper fraction $\frac{29}{24}$. It can be expressed as a mixed number:\n\n$$\frac{29}{24} = 1 \frac{5}{24}$$\n\nSo, the final answer is $\frac{29}{24}$ or $1 \frac{5}{24}$.