Let's solve the expression step by step:

$\left( \frac{8}{125} \right) - \frac{1}{2} + 27 - \frac{1}{3} - 49 \frac{1}{2}$

1. Convert mixed numbers to improper fractions:

   • $49 \frac{1}{2} = 49 + \frac{1}{2} = \frac{99}{2}$

2. Rewrite the expression: $\frac{8}{125} - \frac{1}{2} + 27 - \frac{1}{3} - \frac{99}{2}$

3. Combine like terms:

   • $\frac{8}{125}$ stays as it is.
   • Combine the constants $27 - 49 = -22$.

   Now the expression is: $\frac{8}{125} - \frac{1}{2} - \frac{99}{2} - \frac{1}{3} - 22$

4. Combine the fractions with a common denominator:

   • For $\frac{1}{2}$ and $\frac{99}{2}$: $\frac{1}{2} + \frac{99}{2} = \frac{100}{2} = 50$.

   Now the expression is: $\frac{8}{125} - 50 - \frac{1}{3} - 22$

5. Combine constants:

   • $-50 - 22 = -72$.

   So the expression is: $\frac{8}{125} - \frac{1}{3} - 72$

6. Combine $\frac{8}{125}$ and $\frac{1}{3}$ using a common denominator of 375: $\frac{8}{125} = \frac{24}{375}, \quad \frac{1}{3} = \frac{125}{375}$ Now subtract: $\frac{24}{375} - \frac{125}{375} = \frac{-101}{375}$

7. The final expression is: $\frac{-101}{375} - 72$

8. Convert $\frac{-101}{375}$ to a decimal: $\frac{-101}{375} \approx -0.2693$

9. Add to $-72$: $-72 - 0.2693 = -72.2693$

Thus, the final result is approximately $\boxed{-72.2693}$.

Do you want further details or have any questions?

Here are 5 related questions:

1. How do you simplify mixed numbers to improper fractions?
2. What is the process of finding a common denominator for fractions?
3. How do you convert a fraction to a decimal?
4. How do constants affect the final result in arithmetic expressions?
5. What role does the order of operations play in solving this type of expression?

Tip: When working with fractions and decimals together, converting all terms to decimals can often simplify calculations.