1. The problem is to express the ratio $x : \sqrt{2}$ as an equation with another equivalent ratio. 2. Recall that two ratios $a : b$ and $c : d$ are equal if $\frac{a}{b} = \frac{c}{d}$. 3. Given the ratio $x : \sqrt{2}$, we want to find another ratio $m : n$ such that $$\frac{x}{\sqrt{2}} = \frac{m}{n}$$ 4. To create an equivalent ratio, multiply numerator and denominator by the same nonzero number. For example, multiply both by $\sqrt{2}$: $$\frac{x}{\sqrt{2}} = \frac{x \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{x \sqrt{2}}{2}$$ 5. So the equivalent ratio is $$x : \sqrt{2} = x \sqrt{2} : 2$$ 6. This means the ratio $x : \sqrt{2}$ equals the ratio $x \sqrt{2} : 2$. 7. This is the required form with an equal sign and another ratio.