1. The problem is to simplify the ratio $$\sqrt{12} : \sqrt{300} : \sqrt{48}$$. 2. Recall that the square root of a product can be written as the product of square roots: $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$. 3. Simplify each term by factoring out perfect squares: - $$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$$ - $$\sqrt{300} = \sqrt{100 \times 3} = \sqrt{100} \times \sqrt{3} = 10\sqrt{3}$$ - $$\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$$ 4. Now the ratio is $$2\sqrt{3} : 10\sqrt{3} : 4\sqrt{3}$$. 5. Since all terms have $$\sqrt{3}$$, divide each term by $$\sqrt{3}$$ to simplify: $$2 : 10 : 4$$ 6. Finally, simplify the ratio by dividing all terms by their greatest common divisor, which is 2: $$\frac{2}{2} : \frac{10}{2} : \frac{4}{2} = 1 : 5 : 2$$ 7. Therefore, the simplest form of the ratio is $$1 : 5 : 2$$.