-2/3 · 1/5 - ( 2/3 -1/4) · 2/3 - [ 2/3 · 1/5 - 1/2 : 4]

We need to simplify the given expression step by step:

\[
\frac{-2}{3} \cdot \frac{1}{5} - \left( \frac{2}{3} - \frac{1}{4} \right) \cdot \frac{2}{3} - \left[ \frac{2}{3} \cdot \frac{1}{5} - \frac{1}{2} \div 4 \right]
\]

### Step 1: Simplify individual terms
1. \(\frac{-2}{3} \cdot \frac{1}{5} = \frac{-2}{15}\)
2. Inside the parentheses: 
   - \(\frac{2}{3} - \frac{1}{4} = \frac{8}{12} - \frac{3}{12} = \frac{5}{12}\)
   - Now multiply by \(\frac{2}{3}\):
     \[
     \frac{5}{12} \cdot \frac{2}{3} = \frac{10}{36} = \frac{5}{18}
     \]

### Step 2: Simplify the term inside the brackets
1. \(\frac{2}{3} \cdot \frac{1}{5} = \frac{2}{15}\)
2. \(\frac{1}{2} \div 4 = \frac{1}{2} \cdot \frac{1}{4} = \frac{1}{8}\)

Now, the term inside the brackets becomes:
\[
\frac{2}{15} - \frac{1}{8}
\]
Find the common denominator:
\[
\frac{2}{15} = \frac{16}{120}, \quad \frac{1}{8} = \frac{15}{120}
\]
\[
\frac{2}{15} - \frac{1}{8} = \frac{16}{120} - \frac{15}{120} = \frac{1}{120}
\]

### Step 3: Combine all the results
Now substitute the simplified terms back into the original expression:
\[
\frac{-2}{15} - \frac{5}{18} - \frac{1}{120}
\]
Find a common denominator for all terms (common denominator = 360):
\[
\frac{-2}{15} = \frac{-48}{360}, \quad \frac{5}{18} = \frac{100}{360}, \quad \frac{1}{120} = \frac{3}{360}
\]
Now, sum the fractions:
\[
\frac{-48}{360} - \frac{100}{360} - \frac{3}{360} = \frac{-151}{360}
\]

Thus, the simplified result is:
\[
\boxed{\frac{-151}{360}}
\]

Would you like more details or have any questions?

Here are some related questions to explore:
1. How do you find the least common denominator between fractions?
2. Can you explain why dividing by a fraction is the same as multiplying by its reciprocal?
3. How do you handle negative numbers when multiplying fractions?
4. How does subtracting fractions differ from adding fractions?
5. What are the steps to simplifying complex expressions involving multiple operations?

**Tip:** When working with fractions, always try to simplify each part before combining them into the final expression.

This step-by-step solution simplifies the complex fraction expression: -2/3 · 1/5 - (2/3 - 1/4) · 2/3 - [2/3 · 1/5 - 1/2 ÷ 4]. Learn how to handle multiplication, subtraction, and division of fractions using the least common denominator. Suitable for students in Grades 7-9.

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