1. **State the problem:** Solve the equation $$\frac{n+6}{10} + \frac{2n}{20} = \frac{6}{3n}$$ for $n$. 2. **Identify the formula and rules:** To solve this equation, we need to find a common denominator and clear fractions by multiplying both sides by the least common denominator (LCD). 3. **Find the LCD:** The denominators are 10, 20, and $3n$. The LCD is $60n$. 4. **Multiply both sides by $60n$ to clear denominators:** $$60n \times \frac{n+6}{10} + 60n \times \frac{2n}{20} = 60n \times \frac{6}{3n}$$ 5. **Simplify each term:** - $60n \times \frac{n+6}{10} = 6n(n+6)$ - $60n \times \frac{2n}{20} = 6n \times 2n = 12n^2$ - $60n \times \frac{6}{3n} = 20 \times 6 = 120$ 6. **Rewrite the equation:** $$6n(n+6) + 12n^2 = 120$$ 7. **Expand and simplify:** $$6n^2 + 36n + 12n^2 = 120$$ $$18n^2 + 36n = 120$$ 8. **Bring all terms to one side:** $$18n^2 + 36n - 120 = 0$$ 9. **Divide entire equation by 6 to simplify:** $$3n^2 + 6n - 20 = 0$$ 10. **Use quadratic formula:** $$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=6$, $c=-20$. 11. **Calculate discriminant:** $$\Delta = 6^2 - 4 \times 3 \times (-20) = 36 + 240 = 276$$ 12. **Calculate roots:** $$n = \frac{-6 \pm \sqrt{276}}{6} = \frac{-6 \pm 2\sqrt{69}}{6} = \frac{-3 \pm \sqrt{69}}{3}$$ 13. **Final solutions:** $$n = \frac{-3 + \sqrt{69}}{3} \quad \text{or} \quad n = \frac{-3 - \sqrt{69}}{3}$$