1. The problem is to find the difference between the fractions $\frac{7}{8}$ and $\frac{1}{3}$.\n\n2. To subtract fractions, we need a common denominator. The denominators are 8 and 3. The least common denominator (LCD) is the least common multiple of 8 and 3, which is 24.\n\n3. Convert each fraction to an equivalent fraction with denominator 24:\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 subtract the numerators while keeping the denominator the same:\n$$\frac{21}{24} - \frac{8}{24} = \frac{21 - 8}{24} = \frac{13}{24}$$\n\n5. The fraction $\frac{13}{24}$ cannot be simplified further because 13 is a prime number and does not share any common factors with 24.\n\n6. Therefore, the difference is $\frac{13}{24}$.