1. **State the problem:** Solve for $n$ in the equation $\left(32+\frac{1}{5}\right)-\left(7-\frac{5n}{3}\right)=\frac{5}{6}$.\n\n2. **Rewrite the equation:** \n$$32 + \frac{1}{5} - 7 + \frac{5n}{3} = \frac{5}{6}$$\n\n3. **Simplify constants:** \n$$\left(32 - 7\right) + \frac{1}{5} + \frac{5n}{3} = \frac{5}{6}$$\n$$25 + \frac{1}{5} + \frac{5n}{3} = \frac{5}{6}$$\n\n4. **Combine constants on the left:** \nFind common denominator for $25$ and $\frac{1}{5}$: \n$$25 = \frac{125}{5}$$\nSo, \n$$\frac{125}{5} + \frac{1}{5} = \frac{126}{5}$$\nEquation becomes: \n$$\frac{126}{5} + \frac{5n}{3} = \frac{5}{6}$$\n\n5. **Isolate $\frac{5n}{3}$:** \n$$\frac{5n}{3} = \frac{5}{6} - \frac{126}{5}$$\n\n6. **Find common denominator for right side:** \nCommon denominator is $30$. \n$$\frac{5}{6} = \frac{25}{30}, \quad \frac{126}{5} = \frac{756}{30}$$\nSo, \n$$\frac{5n}{3} = \frac{25}{30} - \frac{756}{30} = -\frac{731}{30}$$\n\n7. **Solve for $n$:** \nMultiply both sides by $\cancel{3}$ and divide by $\cancel{5}$: \n$$n = \frac{-\frac{731}{30} \times \cancel{3}}{\cancel{5}} = -\frac{731}{30} \times \frac{3}{5}$$\n\n8. **Simplify multiplication:** \n$$n = -\frac{731 \times 3}{30 \times 5} = -\frac{2193}{150}$$\n\n9. **Reduce fraction by dividing numerator and denominator by 3:** \n$$n = -\frac{\cancel{2193}^{731}}{\cancel{150}^{50}} = -\frac{731}{50}$$\n\n**Final answer:** \n$$n = -\frac{731}{50}$$