1. **State the problem:** Simplify the expression $$\frac{\frac{12.5}{s(s + 1 + 12.5k)}}{1 + \frac{12.5}{s(s + 1 + 12.5k)}}.$$\n\n2. **Rewrite the expression:** Let $$A = s(s + 1 + 12.5k).$$ Then the expression becomes $$\frac{\frac{12.5}{A}}{1 + \frac{12.5}{A}}.$$\n\n3. **Combine the denominator:** Write the denominator as a single fraction:\n$$1 + \frac{12.5}{A} = \frac{A}{A} + \frac{12.5}{A} = \frac{A + 12.5}{A}.$$\n\n4. **Rewrite the entire expression:**\n$$\frac{\frac{12.5}{A}}{\frac{A + 12.5}{A}}.$$\n\n5. **Divide the fractions:**\n$$= \frac{12.5}{A} \times \frac{A}{A + 12.5}.$$\n\n6. **Cancel common factors:**\n$$= \frac{\cancel{12.5}}{\cancel{A}} \times \frac{\cancel{A}}{A + 12.5} = \frac{12.5}{A + 12.5}.$$\n\n7. **Substitute back for $A$:**\n$$= \frac{12.5}{s(s + 1 + 12.5k) + 12.5}.$$\n\n**Final answer:**\n$$\boxed{\frac{12.5}{s(s + 1 + 12.5k) + 12.5}}.$$