The expression in the image is:

$-3 \frac{2}{5} \div \left( -5 \frac{1}{10} \right)$

We need to first convert the mixed numbers to improper fractions and then perform the division.

Step 1: Convert the mixed numbers to improper fractions.

1. Convert $-3 \frac{2}{5}$: $-3 \frac{2}{5} = -\left( \frac{3 \times 5 + 2}{5} \right) = -\frac{17}{5}$

2. Convert $-5 \frac{1}{10}$: $-5 \frac{1}{10} = -\left( \frac{5 \times 10 + 1}{10} \right) = -\frac{51}{10}$

Step 2: Perform the division of the improper fractions.

$\frac{-17}{5} \div \frac{-51}{10}$

Dividing by a fraction is the same as multiplying by its reciprocal:

$\frac{-17}{5} \times \frac{10}{-51} = \frac{-17 \times 10}{5 \times -51} = \frac{-170}{-255} = \frac{170}{255}$

Step 3: Simplify the fraction.

Find the greatest common divisor (GCD) of 170 and 255. The GCD is 85.

$\frac{170}{255} = \frac{170 \div 85}{255 \div 85} = \frac{2}{3}$

Final Answer:

The result is $\frac{2}{3}$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you convert a mixed number to an improper fraction?
2. Why is dividing by a fraction the same as multiplying by the reciprocal?
3. What steps are involved in simplifying fractions?
4. How can you find the greatest common divisor of two numbers?
5. What other operations can involve fractions, such as addition or subtraction?

Tip: Always simplify mixed numbers to improper fractions before performing operations like multiplication or division for easier calculations.