To solve the given problem, we will break it into parts for clarity. The problem is:

$\left(-\frac{1}{3}\right) - \left\{1 - \left(\frac{2}{3} \div \frac{5}{7}\right)\right\} + 8 - \left[5 - \left(\frac{1}{2} - \frac{1}{4}\right)\right]$

Step-by-Step Solution:

1. Simplify the division inside the curly braces:

$\frac{2}{3} \div \frac{5}{7} = \frac{2}{3} \times \frac{7}{5} = \frac{14}{15}$

2. Subtract this value from 1:

$1 - \frac{14}{15} = \frac{15}{15} - \frac{14}{15} = \frac{1}{15}$

Thus, the expression in the curly braces becomes: $\left\{1 - \left(\frac{2}{3} \div \frac{5}{7}\right)\right\} = \frac{1}{15}$

3. Simplify the subtraction inside the brackets:

$\frac{1}{2} - \frac{1}{4} = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}$

4. Subtract this value from 5:

$5 - \frac{1}{4} = \frac{20}{4} - \frac{1}{4} = \frac{19}{4}$

Thus, the expression in the brackets becomes: $\left[5 - \left(\frac{1}{2} - \frac{1}{4}\right)\right] = \frac{19}{4}$

5. Substitute back into the main expression:

$\left(-\frac{1}{3}\right) - \frac{1}{15} + 8 - \frac{19}{4}$

6. Combine all terms with a common denominator:

The denominators are 3, 15, and 4. The least common denominator (LCD) is 60.

• Convert each term to have a denominator of 60: $-\frac{1}{3} = -\frac{20}{60}, \quad -\frac{1}{15} = -\frac{4}{60}, \quad 8 = \frac{480}{60}, \quad -\frac{19}{4} = -\frac{285}{60}$

• Combine all terms: $-\frac{20}{60} - \frac{4}{60} + \frac{480}{60} - \frac{285}{60} = \frac{-20 - 4 + 480 - 285}{60} = \frac{171}{60}$

7. Simplify the fraction:

$\frac{171}{60} = 2 + \frac{51}{60}$

Thus, the simplified answer is: $\boxed{-5\frac{41}{60}}$

Let me know if you want more clarification or detailed steps.

Related Questions:

1. What is the importance of finding the least common denominator in such problems?
2. How do you perform subtraction with mixed fractions?
3. Can the order of operations change the outcome of a problem? Why or why not?
4. What are some tips for simplifying fractions efficiently?
5. How can these concepts apply to solving real-world problems?

Tip: Always simplify fractions at the end to avoid unnecessary complexity during calculations.