To simplify the expression \((9) \div \left(\frac{5}{4}\right) \times \left(\frac{7}{9}\right) \times (-5) \div \left(\frac{7}{2}\right)\), we will follow the order of operations and apply the properties of multiplication and division.
First, we can rewrite the expression: \[ 9 \div \left(\frac{5}{4}\right) \times \left(\frac{7}{9}\right) \times (-5) \div \left(\frac{7}{2}\right) \]
We'll start by dealing with the divisions. Dividing by a fraction is equal to multiplying by its reciprocal. Thus, we can rewrite \(\div \left(\frac{5}{4}\right)\) and \(\div \left(\frac{7}{2}\right)\) as follows: \[ 9 \times \left(\frac{4}{5}\right) \times \left(\frac{7}{9}\right) \times (-5) \times \left(\frac{2}{7}\right) \]
Now, let's multiply these fractions step-by-step.
-
Combine \(9\) and \(\left(\frac{4}{5}\right)\): \[ 9 \times \left(\frac{4}{5}\right) = \frac{36}{5} \]
-
Multiply by \(\left(\frac{7}{9}\right)\): \[ \frac{36}{5} \times \left(\frac{7}{9}\right) = \frac{36 \times 7}{5 \times 9} = \frac{252}{45} = \frac{28}{5} \text{ (after simplifying)} \]
-
Now multiply by \(-5\): \[ \frac{28}{5} \times (-5) = \frac{28 \times (-5)}{5} = -28 \]
-
Finally, multiply by \(\left(\frac{2}{7}\right)\): \[ -28 \times \left(\frac{2}{7}\right) = \frac{-28 \times 2}{7} = -\frac{56}{7} = -8 \]
Thus, the final answer from the calculation is \(-8\).
The answer is: \[ \boxed{-8} \]