Let's solve the expression step-by-step to ensure clarity and accuracy:

\[ -\frac{2}{5} + \frac{4}{2} \div \frac{3}{2} \cdot \frac{1}{3} + 4 \]

First, we follow the order of operations, often remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

1. **Division and Multiplication**: We need to handle division and multiplication first.
   \[
   \frac{4}{2} \div \frac{3}{2} \cdot \frac{1}{3}
   \]

   Start with the division:
   \[
   \frac{4}{2} \div \frac{3}{2} = \frac{4}{2} \cdot \frac{2}{3} = \frac{4 \cdot 2}{2 \cdot 3} = \frac{8}{6} = \frac{4}{3}
   \]

   Now, multiply the result by \(\frac{1}{3}\):
   \[
   \frac{4}{3} \cdot \frac{1}{3} = \frac{4 \cdot 1}{3 \cdot 3} = \frac{4}{9}
   \]

2. **Addition and Subtraction**: Substitute this result back into the original expression and handle the addition and subtraction:
   \[
   -\frac{2}{5} + \frac{4}{9} + 4
   \]

3. **Combine the fractions**: To combine the fractions \(-\frac{2}{5}\) and \(\frac{4}{9}\), find a common denominator. The least common multiple of 5 and 9 is 45.

   Rewrite the fractions with the common denominator:
   \[
   -\frac{2}{5} = -\frac{2 \cdot 9}{5 \cdot 9} = -\frac{18}{45}
   \]
   \[
   \frac{4}{9} = \frac{4 \cdot 5}{9 \cdot 5} = \frac{20}{45}
   \]

   Add the fractions:
   \[
   -\frac{18}{45} + \frac{20}{45} = \frac{20 - 18}{45} = \frac{2}{45}
   \]

4. **Add 4**: Now, add the integer 4 to the fraction:
   \[
   \frac{2}{45} + 4
   \]

   Convert 4 to a fraction with denominator 45:
   \[
   4 = \frac{4 \cdot 45}{1 \cdot 45} = \frac{180}{45}
   \]

   Add the fractions:
   \[
   \frac{2}{45} + \frac{180}{45} = \frac{2 + 180}{45} = \frac{182}{45}
   \]

Therefore, the value of the expression is:
\[
\frac{182}{45}
\]

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you simplify fractions like \(\frac{182}{45}\)?
2. What are the steps for finding the least common multiple of two numbers?
3. How do you convert a mixed number back into an improper fraction?
4. What are some common mistakes to avoid when following the order of operations?
5. How can you check your work when adding and subtracting fractions?
6. What are the properties of fractions that make them unique compared to whole numbers?
7. How do you handle complex expressions with multiple fractions and operations?
8. How can understanding fractions help with solving real-world problems?

**Tip:** When dealing with fractions, always look for the least common denominator to simplify addition and subtraction.

Learn how to solve the arithmetic expression -2/5 + 4/2 ÷ 3/2 · 1/3 + 4 step-by-step with a detailed explanation. This problem covers fractions, order of operations, and arithmetic operations. Suitable for high school students.

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