Use the properties of operations and rules for multiplying and dividing to solve 8/9 times -3/4 divided 1/9 times -4/5 divided 3/5

To solve the expression \( \frac{8}{9} \times \left(-\frac{3}{4}\right) \div \left(\frac{1}{9} \times \left(-\frac{4}{5}\right) \div \frac{3}{5}\right) \), we will break it down using the properties of operations step by step.

### Step 1: Solve the parts inside the parentheses

**First part**:
\[
\frac{1}{9} \times \left(-\frac{4}{5}\right) = -\frac{4}{45}
\]

**Second part**:
\[
-\frac{4}{45} \div \frac{3}{5}
\]
To divide by a fraction, we multiply by its reciprocal:
\[
-\frac{4}{45} \times \frac{5}{3} = -\frac{4 \cdot 5}{45 \cdot 3} = -\frac{20}{135}
\]
Now, simplify \(-\frac{20}{135}\):
\[
\text{GCD of } 20 \text{ and } 135 = 5
\]
\[
-\frac{20 \div 5}{135 \div 5} = -\frac{4}{27}
\]

### Step 2: Substitute back and evaluate the entire expression

Now we substitute back into the main expression:
\[
\frac{8}{9} \times \left(-\frac{3}{4}\right) \div \left(-\frac{4}{27}\right)
\]

**Multiply \( \frac{8}{9} \) and \( -\frac{3}{4} \)**:
\[
\frac{8 \cdot -3}{9 \cdot 4} = -\frac{24}{36}
\]
Now simplify:
\[
-\frac{24 \div 12}{36 \div 12} = -\frac{2}{3}
\]

**Now divide by \( -\frac{4}{27} \)**:
To divide by a fraction, again we multiply by its reciprocal:
\[
-\frac{2}{3} \div -\frac{4}{27} = -\frac{2}{3} \times -\frac{27}{4} = \frac{2 \cdot 27}{3 \cdot 4} = \frac{54}{12}
\]
Now simplify:
\[
\frac{54 \div 6}{12 \div 6} = \frac{9}{2}
\]

Thus, the final answer is:
\[
\boxed{\frac{9}{2}}
\]

whats it as an mixed number

To convert the improper fraction \(\frac{9}{2}\) into a mixed number, divide the numerator by the denominator.

1. **Division**: \(9 \div 2 = 4\) (which is the whole number part).
2. **Remainder**: \(9 - (2 \times 4) = 9 - 8 = 1\).

So, \(\frac{9}{2}\) can be expressed as:

\[
4\frac{1}{2}
\]

Thus, as a mixed number, \(\frac{9}{2}\) is:

\[
\boxed{4\frac{1}{2}}
\]

wrong whats Use the properties of operations and rules for multiplying and dividing to solve 8/9 times -3/4 divided 1/9 times -4/5 divided 3/5

Let's correctly evaluate the expression

\[
\frac{8}{9} \times \left(-\frac{3}{4}\right) \div \left( \frac{1}{9} \times \left(-\frac{4}{5}\right) \div \frac{3}{5} \right)
\]

### Step 1: Evaluate the inner expression \(\frac{1}{9} \times \left(-\frac{4}{5}\right) \div \frac{3}{5}\)

First, calculate \(\frac{1}{9} \times \left(-\frac{4}{5}\right)\):

\[
\frac{1}{9} \times \left(-\frac{4}{5}\right) = -\frac{4}{45}
\]

Next, divide by \(\frac{3}{5}\):

\[
-\frac{4}{45} \div \frac{3}{5} = -\frac{4}{45} \times \frac{5}{3} = -\frac{4 \cdot 5}{45 \cdot 3} = -\frac{20}{135}
\]

Now simplify \(-\frac{20}{135}\):

The greatest common divisor (GCD) of 20 and 135 is 5:

\[
-\frac{20 \div 5}{135 \div 5} = -\frac{4}{27}
\]

### Step 2: Evaluate the entire expression

Now substitute this value back into the original expression:

\[
\frac{8}{9} \times \left(-\frac{3}{4}\right) \div \left(-\frac{4}{27}\right)
\]

#### Calculate \(\frac{8}{9} \times \left(-\frac{3}{4}\right)\):

\[
\frac{8 \times -3}{9 \times 4} = -\frac{24}{36}
\]

Now simplify \(-\frac{24}{36}\):

The GCD of 24 and 36 is 12:

\[
-\frac{24 \div 12}{36 \div 12} = -\frac{2}{3}
\]

#### Now divide by \(-\frac{4}{27}\):

To divide by a fraction, multiply by its reciprocal:

\[
-\frac{2}{3} \div -\frac{4}{27} = -\frac{2}{3} \times -\frac{27}{4} = \frac{2 \times 27}{3 \times 4} = \frac{54}{12}
\]

Now simplify \(\frac{54}{12}\):

The GCD of 54 and 12 is 6:

\[
\frac{54 \div 6}{12 \div 6} = \frac{9}{2}
\]

### Final Answer

Thus, the resulting value of the expression is:

\[
\boxed{\frac{9}{2}}
\]

To express \(\frac{9}{2}\) as a mixed number:

- \(9 \div 2 = 4\) remainder \(1\).

So, it can be expressed as:

\[
4\frac{1}{2}
\]

Thus, as a mixed number, the answer is:

\[
\boxed{4\frac{1}{2}}
\]