1. **State the problem:** Multiply the fractions $\frac{7}{8}$ and $\frac{5}{6}$. 2. **Formula used:** To multiply fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ 3. **Apply the formula:** $$\frac{7}{8} \times \frac{5}{6} = \frac{7 \times 5}{8 \times 6} = \frac{35}{48}$$ 4. **Simplify if possible:** Check if $35$ and $48$ have common factors. Since $35 = 5 \times 7$ and $48 = 2^4 \times 3$, no common factors exist. So, the fraction is already in simplest form. 5. **Final answer:** $$\boxed{\frac{35}{48}}$$