Let's go through each problem and solve them step by step. Here are the questions as extracted from the image:

Problem (1)

$1 \frac{1}{5} \times \left( -\frac{2}{3} \right) + 1 \frac{1}{5}$

Solution:

1. Convert $1 \frac{1}{5}$ to an improper fraction: $1 \frac{1}{5} = \frac{6}{5}$

2. Substitute and perform the multiplication: $\frac{6}{5} \times -\frac{2}{3} = -\frac{12}{15} = -\frac{4}{5}$

3. Add $1 \frac{1}{5} = \frac{6}{5}$: $-\frac{4}{5} + \frac{6}{5} = \frac{2}{5}$

Answer: $\frac{2}{5}$

Problem (2)

$1 \frac{1}{5} \times (-2) + 1 \frac{1}{5}$

Solution:

1. Convert $1 \frac{1}{5}$ to an improper fraction: $1 \frac{1}{5} = \frac{6}{5}$

2. Multiply: $\frac{6}{5} \times (-2) = -\frac{12}{5}$

3. Add $1 \frac{1}{5} = \frac{6}{5}$: $-\frac{12}{5} + \frac{6}{5} = -\frac{6}{5}$

Answer: $-\frac{6}{5}$

Problem (3)

$\left(-1 \frac{1}{3}\right) \times \left(-2 \frac{1}{4}\right) - 7$

Solution:

1. Convert both mixed numbers to improper fractions: $-1 \frac{1}{3} = -\frac{4}{3} \quad \text{and} \quad -2 \frac{1}{4} = -\frac{9}{4}$

2. Multiply: $-\frac{4}{3} \times -\frac{9}{4} = \frac{36}{12} = 3$

3. Subtract 7: $3 - 7 = -4$

Answer: $-4$

Problem (4)

$-4 \frac{1}{6} - \left(-\frac{1}{2}\right) \times \left(-1 \frac{2}{3}\right)$

Solution:

1. Convert the mixed numbers to improper fractions: $-4 \frac{1}{6} = -\frac{25}{6} \quad \text{and} \quad -1 \frac{2}{3} = -\frac{5}{3}$

2. Perform the multiplication: $-\frac{1}{2} \times -\frac{5}{3} = \frac{5}{6}$

3. Combine terms: $-\frac{25}{6} - \frac{5}{6} = -\frac{30}{6} = -5$

Answer: $-5$

Problem (5)

$\left(-\frac{1}{2}\right) \times (-5) - \left(-1 \frac{1}{2}\right)$

Solution:

1. Convert $-1 \frac{1}{2}$ to an improper fraction: $-1 \frac{1}{2} = -\frac{3}{2}$

2. Perform the multiplication: $-\frac{1}{2} \times -5 = \frac{5}{2}$

3. Subtract $-\frac{3}{2}$: $\frac{5}{2} - (-\frac{3}{2}) = \frac{5}{2} + \frac{3}{2} = \frac{8}{2} = 4$

Answer: $4$

Would you like a deeper explanation of any specific step? Here are five related questions you might find useful:

1. How can you simplify calculations when working with mixed numbers?
2. What is the process for converting between improper fractions and mixed numbers?
3. How do negative signs impact multiplication with fractions?
4. How does the order of operations affect calculations with multiple steps?
5. What is the best method for simplifying fractions after multiplication?

Tip: Always convert mixed numbers to improper fractions before performing operations. This reduces errors and makes calculations simpler.