1. **State the problem:** Simplify the expression $$\frac{\frac{u}{5}}{\frac{25}{u-5} + \frac{u-5}{25}}$$. 2. **Rewrite the denominator:** The denominator is a sum of two fractions: $$\frac{25}{u-5} + \frac{u-5}{25}$$ 3. **Find a common denominator for the denominator's fractions:** The common denominator is $$25(u-5)$$. 4. **Rewrite each fraction with the common denominator:** $$\frac{25}{u-5} = \frac{25 \times 25}{(u-5) \times 25} = \frac{625}{25(u-5)}$$ $$\frac{u-5}{25} = \frac{(u-5) \times (u-5)}{25 \times (u-5)} = \frac{(u-5)^2}{25(u-5)}$$ 5. **Add the fractions in the denominator:** $$\frac{625}{25(u-5)} + \frac{(u-5)^2}{25(u-5)} = \frac{625 + (u-5)^2}{25(u-5)}$$ 6. **Rewrite the original expression:** $$\frac{\frac{u}{5}}{\frac{625 + (u-5)^2}{25(u-5)}} = \frac{u}{5} \times \frac{25(u-5)}{625 + (u-5)^2}$$ 7. **Simplify the multiplication:** $$= \frac{u \times 25 (u-5)}{5 \times [625 + (u-5)^2]}$$ 8. **Cancel common factors:** $$= \frac{u \times \cancel{25} (u-5)}{\cancel{5} \times [625 + (u-5)^2]} = \frac{u \times 5 (u-5)}{625 + (u-5)^2}$$ 9. **Expand numerator:** $$5u(u-5) = 5u^2 - 25u$$ 10. **Expand denominator term:** $$(u-5)^2 = u^2 - 10u + 25$$ 11. **Sum denominator:** $$625 + u^2 - 10u + 25 = u^2 - 10u + 650$$ 12. **Final simplified expression:** $$\frac{5u^2 - 25u}{u^2 - 10u + 650}$$ **Answer:** $$\boxed{\frac{5u^2 - 25u}{u^2 - 10u + 650}}$$