Let's analyze and solve the given mathematical expression step by step. The problem is:

$5 \cdot \left(\frac{8}{5} - \frac{4}{7}\right) \div \frac{9}{7} - 3 \cdot \left(4 - \frac{1}{5}\right) \cdot \frac{1}{19}$

Step 1: Simplify inside the parentheses

1.1 Simplify $\frac{8}{5} - \frac{4}{7}$:

Find the common denominator: $\text{Common denominator} = 35.$

Rewriting the fractions: $\frac{8}{5} = \frac{56}{35}, \quad \frac{4}{7} = \frac{20}{35}.$

Subtract: $\frac{8}{5} - \frac{4}{7} = \frac{56}{35} - \frac{20}{35} = \frac{36}{35}.$

1.2 Simplify $4 - \frac{1}{5}$:

Convert $4$ into a fraction with denominator 5: $4 = \frac{20}{5}.$

Subtract: $4 - \frac{1}{5} = \frac{20}{5} - \frac{1}{5} = \frac{19}{5}.$

Step 2: Divide and multiply terms

2.1 Simplify $\left(\frac{8}{5} - \frac{4}{7}\right) \div \frac{9}{7}$:

Division by a fraction is the same as multiplying by its reciprocal: $\frac{36}{35} \div \frac{9}{7} = \frac{36}{35} \cdot \frac{7}{9}.$

Simplify the multiplication: $\frac{36 \cdot 7}{35 \cdot 9} = \frac{252}{315}.$

Simplify the fraction: $\frac{252}{315} = \frac{4}{5}.$

2.2 Simplify $3 \cdot \left(4 - \frac{1}{5}\right) \cdot \frac{1}{19}$:

Substitute $4 - \frac{1}{5} = \frac{19}{5}$: $3 \cdot \frac{19}{5} \cdot \frac{1}{19}.$

Cancel out $19$ in the numerator and denominator: $3 \cdot \frac{1}{5} = \frac{3}{5}.$

Step 3: Final computation

Substitute back: $5 \cdot \frac{4}{5} - \frac{3}{5}.$

3.1 Simplify $5 \cdot \frac{4}{5}$:

$5 \cdot \frac{4}{5} = 4.$

3.2 Compute $4 - \frac{3}{5}$:

Convert $4$ into a fraction with denominator 5: $4 = \frac{20}{5}.$

Subtract: $\frac{20}{5} - \frac{3}{5} = \frac{17}{5}.$

Final Answer:

$\boxed{\frac{17}{5}}$

Let me know if you need additional details or clarification!

Related Questions:

1. How do you find a common denominator for fractions?
2. How do you simplify complex fractions involving division and multiplication?
3. What is the process for converting mixed numbers into improper fractions?
4. How do you handle expressions with multiple parentheses?
5. How can you verify the correctness of fraction operations?

Tip:

Always simplify fractions at intermediate steps to avoid large numbers and make calculations easier!