1. **State the problem:** Simplify the ratio $$\sqrt{12} : \sqrt{300} : \sqrt{48}$$ to its simplest form. 2. **Recall the rule:** The ratio of square roots can be simplified by expressing each square root in terms of its prime factors and then simplifying. 3. **Simplify each term:** - $$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$$ - $$\sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3}$$ - $$\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$$ 4. **Rewrite the ratio:** $$2\sqrt{3} : 10\sqrt{3} : 4\sqrt{3}$$ 5. **Cancel the common factor $$\sqrt{3}$$:** $$2 : 10 : 4$$ 6. **Simplify the ratio by dividing each term by the greatest common divisor (GCD) of 2, 10, and 4, which is 2:** $$\frac{2}{2} : \frac{10}{2} : \frac{4}{2} = 1 : 5 : 2$$ **Final answer:** $$1 : 5 : 2$$