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To solve the given expression step by step, we'll first handle each part of the equation and then solve it systematically.

The given expression is:

\[
\frac{4}{3} + \frac{7}{5} \cdot \left( \frac{1}{2} + \frac{4}{9} \right) - \frac{1}{5}
\]

### Step 1: Simplify the expression inside the parentheses
First, add the fractions inside the parentheses:
\[
\frac{1}{2} + \frac{4}{9}
\]

To add these fractions, find the least common denominator (LCD). The denominators are 2 and 9, and the LCD is 18.

\[
\frac{1}{2} = \frac{9}{18}, \quad \frac{4}{9} = \frac{8}{18}
\]
\[
\frac{9}{18} + \frac{8}{18} = \frac{17}{18}
\]

### Step 2: Multiply the result with \(\frac{7}{5}\)
Next, multiply \(\frac{7}{5}\) by the result from Step 1:
\[
\frac{7}{5} \cdot \frac{17}{18} = \frac{119}{90}
\]

### Step 3: Add \(\frac{4}{3}\) to the result
Now, add \(\frac{4}{3}\) to \(\frac{119}{90}\). First, find the least common denominator for 3 and 90, which is 90.

\[
\frac{4}{3} = \frac{120}{90}
\]

So,
\[
\frac{120}{90} + \frac{119}{90} = \frac{239}{90}
\]

### Step 4: Subtract \(\frac{1}{5}\) from the result
Finally, subtract \(\frac{1}{5}\) from \(\frac{239}{90}\). Convert \(\frac{1}{5}\) to have the same denominator as 90.

\[
\frac{1}{5} = \frac{18}{90}
\]
\[
\frac{239}{90} - \frac{18}{90} = \frac{221}{90}
\]

### Final Answer
The final result is:

\[
\boxed{\frac{221}{90}}
\]

Would you like more details or have any other questions? Here are five follow-up questions:

1. How do you find the least common denominator of multiple fractions?
2. What steps should be taken when dealing with parentheses in an expression?
3. How can improper fractions be converted into mixed numbers?
4. Why is it important to simplify an expression step by step?
5. How can this process be applied to solve more complex algebraic expressions?

**Tip:** Always work with common denominators when adding or subtracting fractions to simplify calculations and avoid mistakes.

Learn how to solve a mixed fraction problem step-by-step. This problem involves operations on fractions, including addition, multiplication, and subtraction. Suitable for students in Grades 5-6.

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