Let's evaluate the given expression step by step. The expression is:

$\frac{1}{5} \div \left(-\frac{1}{2} \times \frac{3}{4}\right) + \frac{4}{5}$

Step 1: Evaluate the multiplication $-\frac{1}{2} \times \frac{3}{4}$

$-\frac{1}{2} \times \frac{3}{4} = -\frac{1 \times 3}{2 \times 4} = -\frac{3}{8}$

Step 2: Evaluate the division $\frac{1}{5} \div -\frac{3}{8}$

Dividing by a fraction is the same as multiplying by its reciprocal: $\frac{1}{5} \div -\frac{3}{8} = \frac{1}{5} \times -\frac{8}{3} = -\frac{1 \times 8}{5 \times 3} = -\frac{8}{15}$

Step 3: Add $-\frac{8}{15}$ to $\frac{4}{5}$

To add these fractions, we need a common denominator. The least common denominator between 15 and 5 is 15. Convert $\frac{4}{5}$ to a fraction with a denominator of 15: $\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}$ Now add the fractions: $-\frac{8}{15} + \frac{12}{15} = \frac{-8 + 12}{15} = \frac{4}{15}$

Final Answer:

The value of the expression is $\frac{4}{15}$.

Would you like further details or have any questions? Here are some related questions:

1. How do you multiply and divide fractions with negative signs?
2. What is the importance of finding a common denominator when adding or subtracting fractions?
3. How do you convert a fraction into a mixed number?
4. What are the steps for simplifying fractions?
5. How do you handle complex fractions with multiple operations?

Tip: When performing operations on fractions, always keep track of the signs and ensure you follow the order of operations (PEMDAS).