To solve the expression \((-4) \times \left(\frac{7}{5}\right) \times \left(-\frac{3}{4}\right) \div (7 \frac{1}{5})\), we can break it down step by step, applying the properties of operations for multiplying and dividing signed numbers.
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Convert the mixed number into an improper fraction: \[ 7 \frac{1}{5} = 7 + \frac{1}{5} = \frac{35}{5} + \frac{1}{5} = \frac{36}{5} \]
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Re-write the expression: \[ (-4) \times \left(\frac{7}{5}\right) \times \left(-\frac{3}{4}\right) \div \left(\frac{36}{5}\right) \]
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Replace the division with multiplication by the reciprocal: \[ (-4) \times \left(\frac{7}{5}\right) \times \left(-\frac{3}{4}\right) \times \left(\frac{5}{36}\right) \]
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Multiply the numbers:
- First, combine the signs: \((-4) \times (-\frac{3}{4}) = 3\).
- Now we will multiply everything else: \[ 3 \times \left(\frac{7}{5}\right) \times \left(\frac{5}{36}\right) \]
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Combine the fractions: \[ 3 \times \frac{7 \times 5}{5 \times 36} = \frac{3 \times 7}{36} = \frac{21}{36} \]
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Simplify the fraction: \[ \frac{21}{36} = \frac{7}{12} \]
The final answer is: \[ \frac{7}{12} \]