Let's break down and solve the expression step by step:

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

### Step 1: Simplify \(\frac{2}{3} - \frac{1}{3} \times \frac{7}{2}\)
First, calculate the multiplication:

\[
\frac{1}{3} \times \frac{7}{2} = \frac{7}{6}
\]

Now subtract:

\[
\frac{2}{3} - \frac{7}{6} = \frac{4}{6} - \frac{7}{6} = \frac{-3}{6} = -\frac{1}{2}
\]

### Step 2: Simplify \(\frac{1}{2} \times \left(-\frac{2}{3}\right)\)
\[
\frac{1}{2} \times \left(-\frac{2}{3}\right) = -\frac{1}{3}
\]

### Step 3: Simplify \(\frac{1}{3} \div \frac{2}{5}\)
Dividing by a fraction is the same as multiplying by its reciprocal:

\[
\frac{1}{3} \div \frac{2}{5} = \frac{1}{3} \times \frac{5}{2} = \frac{5}{6}
\]

### Step 4: Combine all parts
Now we add up the simplified results from each step:

\[
-\frac{1}{2} + \left(-\frac{1}{3}\right) + \frac{5}{6}
\]

To add these fractions, convert them to have a common denominator (6):

\[
-\frac{1}{2} = -\frac{3}{6}, \quad -\frac{1}{3} = -\frac{2}{6}, \quad \frac{5}{6} = \frac{5}{6}
\]

Now add:

\[
-\frac{3}{6} + \left(-\frac{2}{6}\right) + \frac{5}{6} = \frac{-3 - 2 + 5}{6} = \frac{0}{6} = 0
\]

### Final Answer: 
The value of the expression is \(0\).

Would you like more details on any step?

Here are 5 questions to explore further:
1. What is the general process for simplifying expressions with fractions?
2. How do you handle multiplication and division of fractions?
3. Why is it important to find a common denominator when adding or subtracting fractions?
4. How do you convert a division of fractions into a multiplication?
5. How can mixed numbers be simplified in more complex expressions?

**Tip:** When working with multiple fractions in an expression, always perform the operations in the correct order (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction).

Learn how to simplify the expression (2/3 - 1/3 * 7/2) + 1/2(-2/3) + 1/3 ÷ 2/5 with a detailed step-by-step solution. This problem covers key mathematical concepts like fractions and the order of operations, suitable for students in Grades 6-8.

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