1. **State the problem:** Simplify the ratio $$\frac{25w^4}{45w^7}$$. 2. **Recall the rules:** - When dividing coefficients, divide the numbers normally. - When dividing variables with exponents, subtract the exponents: $$\frac{w^a}{w^b} = w^{a-b}$$. 3. **Simplify the coefficients:** $$\frac{25}{45} = \frac{5 \times 5}{5 \times 9} = \frac{5}{9}$$. 4. **Simplify the variables:** $$w^{4-7} = w^{-3}$$. 5. **Combine the simplified parts:** $$\frac{5}{9} w^{-3}$$. 6. **Rewrite with positive exponents:** $$w^{-3} = \frac{1}{w^3}$$, so $$\frac{5}{9} w^{-3} = \frac{5}{9 w^3}$$. **Final answer:** $$\boxed{\frac{5}{9 w^3}}$$