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To solve the given expression, let's first rewrite it clearly and then solve it step by step.

### Expression:
\[
\left(\frac{3}{4} + 1\right) \div \left(\frac{5}{4} + 2\right) - \left(9 - 3\right) - \left(\left(\frac{5}{9} \div \frac{12}{3}\right)\right)
\]

### Step 1: Simplify the terms inside the parentheses.
- For the first part:
  \[
  \frac{3}{4} + 1 = \frac{3}{4} + \frac{4}{4} = \frac{7}{4}
  \]
  So the expression now becomes:
  \[
  \frac{7}{4} \div \left(\frac{5}{4} + 2\right)
  \]

- For the second part:
  \[
  \frac{5}{4} + 2 = \frac{5}{4} + \frac{8}{4} = \frac{13}{4}
  \]
  So the expression now becomes:
  \[
  \frac{7}{4} \div \frac{13}{4}
  \]

### Step 2: Simplify the division of fractions.
To divide \(\frac{7}{4}\) by \(\frac{13}{4}\), we multiply by the reciprocal of \(\frac{13}{4}\):
\[
\frac{7}{4} \div \frac{13}{4} = \frac{7}{4} \times \frac{4}{13} = \frac{28}{52} = \frac{7}{13}
\]

So, the first part of the expression simplifies to \(\frac{7}{13}\).

### Step 3: Simplify the middle part \((9 - 3)\):
\[
9 - 3 = 6
\]

### Step 4: Simplify the third part \(\left(\frac{5}{9} \div \frac{12}{3}\right)\):
First, simplify the division within the parentheses:
\[
\frac{12}{3} = 4
\]
\[
\frac{5}{9} \div 4 = \frac{5}{9} \times \frac{1}{4} = \frac{5}{36}
\]

### Step 5: Combine all the simplified parts:
Now, the expression becomes:
\[
\frac{7}{13} - 6 - \frac{5}{36}
\]

### Step 6: Simplify further by converting everything to a common denominator:
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 13 and 36 is 468.

Convert each fraction:
\[
\frac{7}{13} = \frac{252}{468}, \quad \frac{5}{36} = \frac{65}{468}
\]
So:
\[
\frac{252}{468} - 6 - \frac{65}{468} = \frac{252 - 65}{468} - 6 = \frac{187}{468} - 6
\]
Now, we need to subtract 6 from the fraction. Convert 6 to a fraction with a denominator of 468:
\[
6 = \frac{2808}{468}
\]
Finally:
\[
\frac{187}{468} - \frac{2808}{468} = \frac{187 - 2808}{468} = \frac{-2621}{468}
\]

### Final Answer:
The expression simplifies to \(\frac{-2621}{468}\), which is a negative improper fraction.

Would you like more details on any specific step or have further questions?

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### Relative Questions:
1. What is the process for finding the LCM of two numbers?
2. How do you multiply and divide fractions?
3. How can you convert a mixed number into an improper fraction?
4. What are some common errors to avoid when subtracting fractions?
5. How do you interpret the final fraction in terms of its decimal equivalent?

### Tip:
Always simplify fractions at every step to avoid large numbers in final calculations.

Learn how to simplify the complex fraction expression (3/4 + 1) / (5/4 + 2) - (9 - 3) - ((5/9) ÷ (12/3)) step by step. This problem covers fraction simplification, division of fractions, and arithmetic operations. Suitable for students in Grades 7-9.

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