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The expression you provided is:

\[
\frac{6}{5} \left( 1 - \frac{2}{7} \right) : \frac{7}{9} \left( \frac{81}{21} \right) + \frac{3}{4} \cdot \frac{6}{9}
\]

I'll go ahead and solve it step-by-step:

### Step 1: Simplify the first term
\[
\frac{6}{5} \left( 1 - \frac{2}{7} \right)
\]
First, simplify inside the parentheses:
\[
1 - \frac{2}{7} = \frac{7}{7} - \frac{2}{7} = \frac{5}{7}
\]
Now multiply:
\[
\frac{6}{5} \times \frac{5}{7} = \frac{6 \times 5}{5 \times 7} = \frac{30}{35} = \frac{6}{7}
\]

### Step 2: Simplify the second term
\[
\frac{7}{9} \times \frac{81}{21}
\]
First, simplify \(\frac{81}{21}\):
\[
\frac{81}{21} = \frac{81 \div 3}{21 \div 3} = \frac{27}{7}
\]
Now multiply:
\[
\frac{7}{9} \times \frac{27}{7} = \frac{7 \times 27}{9 \times 7} = \frac{27}{9} = 3
\]

### Step 3: Solve the division
The original expression had a division symbol between the first and second terms:
\[
\frac{6}{7} : 3 = \frac{6}{7} \times \frac{1}{3} = \frac{6}{7 \times 3} = \frac{6}{21} = \frac{2}{7}
\]

### Step 4: Simplify the third term
\[
\frac{3}{4} \times \frac{6}{9}
\]
Simplify \(\frac{6}{9}\):
\[
\frac{6}{9} = \frac{2}{3}
\]
Now multiply:
\[
\frac{3}{4} \times \frac{2}{3} = \frac{3 \times 2}{4 \times 3} = \frac{6}{12} = \frac{1}{2}
\]

### Step 5: Add the results
Now add the results of the division and the multiplication:
\[
\frac{2}{7} + \frac{1}{2}
\]
To add these fractions, we need a common denominator. The least common denominator between 7 and 2 is 14:
\[
\frac{2}{7} = \frac{4}{14}, \quad \frac{1}{2} = \frac{7}{14}
\]
Now add:
\[
\frac{4}{14} + \frac{7}{14} = \frac{11}{14}
\]

### Final Answer:
\[
\boxed{\frac{11}{14}}
\]

Would you like any more details or have any questions? 

Here are 5 related questions you could explore:
1. How do you simplify complex fractions involving both division and multiplication?
2. What are the rules for dividing fractions?
3. How can you find a common denominator quickly when adding fractions?
4. How can the distributive property help in simplifying similar expressions?
5. How would you solve a more complex algebraic fraction with variables?

**Tip**: Always simplify fractions early in the process to make calculations easier.

6/5 (1 - 2/7) : 7/9 (81/21) + 3/4 · 6/9 = ?

Learn how to solve a complex fraction problem step-by-step. This problem involves division, multiplication, and addition of fractions, covering key arithmetic operations with fractions. Suitable for students in Grades 6-8.

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