1. The problem is to find the missing numerator in the equation $$\frac{?}{10 r w^2} = \frac{2 r t}{5 w}$$. 2. To solve for the missing numerator, we use the property of equality of fractions: if $$\frac{a}{b} = \frac{c}{d}$$, then $$a \times d = b \times c$$. 3. Applying this to our equation, let the missing numerator be $x$. Then: $$x \times 5 w = 10 r w^2 \times 2 r t$$ 4. Simplify the right side: $$10 r w^2 \times 2 r t = 20 r^2 w^2 t$$ 5. So the equation becomes: $$5 w x = 20 r^2 w^2 t$$ 6. To isolate $x$, divide both sides by $5 w$: $$x = \frac{20 r^2 w^2 t}{5 w}$$ 7. Cancel common factors $5$ and $w$: $$x = \frac{\cancel{20}^{4} r^2 \cancel{w^2}^{w} t}{\cancel{5} \cancel{w}} = 4 r^2 w t$$ 8. Therefore, the missing numerator is: $$\boxed{4 r^2 w t}$$