Let's break down the operation:

$- \frac{2}{5} - 3\frac{1}{2}$

Step 1: Convert the mixed number to an improper fraction.

For $3\frac{1}{2}$, convert the whole number and fraction into a single improper fraction:

$3\frac{1}{2} = \frac{(3 \times 2) + 1}{2} = \frac{7}{2}$

Now, the expression becomes:

$- \frac{2}{5} - \frac{7}{2}$

Step 2: Find a common denominator.

The denominators are 5 and 2. The least common denominator (LCD) is 10. Rewrite both fractions with denominator 10:

$- \frac{2}{5} = - \frac{4}{10}, \quad \frac{7}{2} = \frac{35}{10}$

Now the expression is:

$- \frac{4}{10} - \frac{35}{10}$

Step 3: Combine the fractions.

Since the denominators are the same, combine the numerators:

$- \frac{4 + 35}{10} = - \frac{39}{10}$

Step 4: Convert the improper fraction to a mixed number.

To convert $- \frac{39}{10}$ into a mixed number:

$39 \div 10 = 3 \text{ remainder } 9$

Thus:

$- \frac{39}{10} = - 3\frac{9}{10}$

Final Answer:

$- 3\frac{9}{10}$

Let me know if you need further clarification or details!

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Here are 5 related questions to explore further:

1. How do you convert between improper fractions and mixed numbers?
2. What is the least common denominator (LCD), and how do you find it?
3. How do you simplify a fraction before performing addition or subtraction?
4. What is the process for adding and subtracting fractions with unlike denominators?
5. How do you handle negative signs in fraction operations?

Tip: Always convert mixed numbers to improper fractions before performing arithmetic operations like addition or subtraction.