1. **State the problem:** Solve for $n$ in the equation $$\frac{3n+1}{5} - \frac{7-5n}{3} = \frac{5}{6}.$$\n\n2. **Identify the formula and rules:** To solve equations with fractions, find a common denominator to clear fractions by multiplying both sides. The least common denominator (LCD) of 5, 3, and 6 is 30.\n\n3. **Multiply both sides by 30 to clear denominators:**\n$$30 \times \left(\frac{3n+1}{5} - \frac{7-5n}{3}\right) = 30 \times \frac{5}{6}$$\n\n4. **Simplify each term:**\n$$30 \times \frac{3n+1}{5} = 6(3n+1) = 18n + 6$$\n$$30 \times \frac{7-5n}{3} = 10(7-5n) = 70 - 50n$$\n$$30 \times \frac{5}{6} = 5 \times 5 = 25$$\n\n5. **Rewrite the equation:**\n$$18n + 6 - (70 - 50n) = 25$$\n\n6. **Distribute the minus sign:**\n$$18n + 6 - 70 + 50n = 25$$\n\n7. **Combine like terms:**\n$$18n + 50n + 6 - 70 = 25$$\n$$68n - 64 = 25$$\n\n8. **Add 64 to both sides:**\n$$68n - 64 + 64 = 25 + 64$$\n$$68n = 89$$\n\n9. **Divide both sides by 68:**\n$$n = \frac{\cancel{68}n}{\cancel{68}} = \frac{89}{68}$$\n\n**Final answer:** $$n = \frac{89}{68}.$$