1. The problem is to simplify and evaluate the expression: $$\frac{314 \times (1.96)^2 \times 0.613 \times (1-0.613)}{(0.1)^2 \times (314-1) + 314 \times (1.96)^2 \times 0.613 \times (1-0.613)}$$ 2. First, calculate the components step-by-step. 3. Calculate $1-0.613 = 0.387$. 4. Calculate $(1.96)^2 = 3.8416$. 5. Calculate numerator: $$314 \times 3.8416 \times 0.613 \times 0.387$$ 6. Calculate denominator: $$(0.1)^2 \times (314-1) + 314 \times 3.8416 \times 0.613 \times 0.387$$ 7. Calculate $(0.1)^2 = 0.01$. 8. Calculate $0.01 \times 313 = 3.13$. 9. Calculate numerator value: $$314 \times 3.8416 \times 0.613 \times 0.387 \approx 286.3$$ 10. Calculate denominator value: $$3.13 + 286.3 = 289.43$$ 11. Finally, calculate the fraction: $$\frac{286.3}{289.43} \approx 0.989$$ 12. So, the value of the expression is approximately $0.989$.