1. The problem is to evaluate the expression $11 \frac{6}{45}$. 2. First, convert the mixed number to an improper fraction. The whole number is 11 and the fraction is $\frac{6}{45}$. 3. The improper fraction form is $$11 + \frac{6}{45} = \frac{11 \times 45}{45} + \frac{6}{45} = \frac{495}{45} + \frac{6}{45} = \frac{495 + 6}{45} = \frac{501}{45}.$$ 4. Simplify the fraction $\frac{501}{45}$. Find the greatest common divisor (GCD) of 501 and 45. 5. The prime factors of 501 are $3 \times 167$, and for 45 are $3^2 \times 5$. The common factor is 3. 6. Divide numerator and denominator by 3: $$\frac{\cancel{3} \times 167}{\cancel{3} \times 15} = \frac{167}{15}.$$ 7. Convert back to a mixed number: $$167 \div 15 = 11 \text{ remainder } 2,$$ so $$\frac{167}{15} = 11 \frac{2}{15}.$$ 8. Therefore, the simplified value of $11 \frac{6}{45}$ is $11 \frac{2}{15}$.