1. **State the problem:** Solve the equation $$12 \times \frac{120}{n + 5} = 12 \times \frac{120}{n} - 24$$ for $n$. 2. **Write the equation clearly:** $$12 \cdot \frac{120}{n + 5} = 12 \cdot \frac{120}{n} - 24$$ 3. **Simplify the terms:** $$\frac{1440}{n + 5} = \frac{1440}{n} - 24$$ 4. **Isolate the fractions:** Move all terms to one side: $$\frac{1440}{n + 5} - \frac{1440}{n} = -24$$ 5. **Find common denominator and combine:** $$\frac{1440n - 1440(n + 5)}{n(n + 5)} = -24$$ 6. **Simplify numerator:** $$1440n - 1440n - 7200 = -7200$$ So, $$\frac{-7200}{n(n + 5)} = -24$$ 7. **Multiply both sides by $n(n + 5)$:** $$-7200 = -24 \times n(n + 5)$$ 8. **Divide both sides by -24:** $$\cancel{-7200} \div \cancel{-24} = n(n + 5)$$ $$300 = n^2 + 5n$$ 9. **Rewrite as quadratic equation:** $$n^2 + 5n - 300 = 0$$ 10. **Solve quadratic using the quadratic formula:** $$n = \frac{-5 \pm \sqrt{5^2 - 4 \times 1 \times (-300)}}{2 \times 1} = \frac{-5 \pm \sqrt{25 + 1200}}{2} = \frac{-5 \pm \sqrt{1225}}{2}$$ 11. **Calculate the square root:** $$\sqrt{1225} = 35$$ 12. **Find the two solutions:** $$n = \frac{-5 + 35}{2} = \frac{30}{2} = 15$$ $$n = \frac{-5 - 35}{2} = \frac{-40}{2} = -20$$ 13. **Check for restrictions:** Denominators $n$ and $n+5$ cannot be zero, so $n \neq 0$ and $n \neq -5$. Both 15 and -20 are valid. **Final answer:** $$n = 15 \text{ or } n = -20$$