To solve the expression \((9) \div (5/4) \times (7/9) \times (-5) \div (7/2)\), we can follow the order of operations and simplify step by step.
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Rewrite the division as multiplication by the reciprocal: \[ (9) \div (5/4) = 9 \times (4/5) \] Therefore, \[ 9 \div (5/4) = 9 \times (4/5) = \frac{9 \times 4}{5} = \frac{36}{5} \]
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Now, substitute this back into the expression: \[ \frac{36}{5} \times (7/9) \times (-5) \div (7/2) \]
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Rewrite the division by \((7/2)\) as multiplication by its reciprocal: \[ (-5) \div (7/2) = (-5) \times (2/7) \]
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Combining everything, we now have: \[ \frac{36}{5} \times \frac{7}{9} \times (-5) \times \frac{2}{7} \]
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Now, we can combine these fractions: \[ \frac{36}{5} \times \frac{7}{9} \times \frac{-5}{1} \times \frac{2}{7} \]
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Notice that \(\frac{7}{7}\) cancels out, simplifying the expression: \[ = \frac{36}{5} \times \frac{-5}{9} \times 2 \]
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Now, simplify this step-by-step:
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Multiply \(\frac{36}{5}\) by \(-5\): \[ = \frac{36 \cdot -5}{5} = -36 \]
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Then multiply \(-36\) by \(\frac{2}{9}\): \[ -36 \times \frac{2}{9} = \frac{-72}{9} = -8 \]
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So, the final answer is: \[ \boxed{-8} \]