To simplify the expression \(\frac{9}{\frac{5}{4}} \times \frac{7}{9} \times \frac{-5}{\frac{7}{2}}\), we can break it down step by step.
-
Simplify each part:
- The expression \(\frac{9}{\frac{5}{4}}\) can be rewritten as \(9 \times \frac{4}{5} = \frac{36}{5}\).
- The fraction \(-\frac{5}{\frac{7}{2}}\) can be rewritten as \(-5 \times \frac{2}{7} = -\frac{10}{7}\).
-
Rewrite the full expression: Now the expression becomes: \[ \frac{36}{5} \times \frac{7}{9} \times \left(-\frac{10}{7}\right) \]
-
Multiply the fractions: To multiply the fractions, we multiply the numerators together and the denominators together: \[ \frac{36 \times 7 \times -10}{5 \times 9 \times 7} \]
We can simplify:
- The 7 in the numerator and the 7 in the denominator cancel each other out.
-
Now we have: \[ \frac{36 \times -10}{5 \times 9} \]
-
Calculate the numerator and denominator separately:
- Numerator: \(36 \times -10 = -360\)
- Denominator: \(5 \times 9 = 45\)
-
We have: \[ \frac{-360}{45} \]
-
Simplify: Divide both the numerator and the denominator by 45: \[ -360 \div 45 = -8 \]
Thus, the final answer is: \[ \boxed{-8} \]