1. **State the problem:** Solve for $n$ in the equation $$\frac{5}{3} = \frac{5}{n+2} - \frac{n}{6n}.$$\n\n2. **Rewrite the equation:** Note that $\frac{n}{6n} = \frac{1}{6}$ for $n \neq 0$. So the equation becomes $$\frac{5}{3} = \frac{5}{n+2} - \frac{1}{6}.$$\n\n3. **Isolate the fraction with $n$:** Add $\frac{1}{6}$ to both sides: $$\frac{5}{3} + \frac{1}{6} = \frac{5}{n+2}.$$\n\n4. **Find common denominator on the left:** The common denominator of 3 and 6 is 6, so $$\frac{5}{3} = \frac{10}{6}.$$ Thus, $$\frac{10}{6} + \frac{1}{6} = \frac{11}{6}.$$\n\n5. **Rewrite the equation:** $$\frac{11}{6} = \frac{5}{n+2}.$$\n\n6. **Cross multiply:** $$11(n+2) = 6 \times 5.$$\n\n7. **Simplify:** $$11n + 22 = 30.$$\n\n8. **Solve for $n$:** Subtract 22 from both sides: $$11n = 8.$$ Divide both sides by 11: $$n = \frac{8}{11}.$$\n\n**Final answer:** $$n = \frac{8}{11}.$$