To solve the expression \((9) \div \left(\frac{5}{4} \times \frac{7}{9} \times (-5)\right) \div \left(\frac{7}{2}\right)\), we'll step through it carefully.
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Calculate the product in the denominator: \[ \frac{5}{4} \times \frac{7}{9} \times (-5) \] First, multiply \(\frac{5}{4}\) and \(\frac{7}{9}\): \[ \frac{5 \times 7}{4 \times 9} = \frac{35}{36} \] Now, multiply by \(-5\): \[ \frac{35}{36} \times (-5) = \frac{35 \times (-5)}{36} = \frac{-175}{36} \]
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Divide by \(\frac{7}{2}\): To divide by \(\frac{7}{2}\), we multiply by its reciprocal: \[ \frac{-175}{36} \div \frac{7}{2} = \frac{-175}{36} \times \frac{2}{7} = \frac{-175 \times 2}{36 \times 7} = \frac{-350}{252} \]
We can simplify \(\frac{-350}{252}\):
- Divide both the numerator and the denominator by their greatest common divisor, which is 14. \[ \frac{-350 \div 14}{252 \div 14} = \frac{-25}{18} \]
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Now divide 9 by this result: \[ 9 \div \left(\frac{-25}{18}\right) = 9 \times \left(\frac{-18}{25}\right) = \frac{9 \times -18}{25} = \frac{-162}{25} \]
Thus, the final result is: \[ \boxed{\frac{-162}{25}} \]