To solve the expression \((-2) \times \left(\frac{1}{5}\right) \div \left(\frac{2}{3}\right) \times \left(-\frac{5}{6}\right) \div \left(\frac{3}{4}\right)\), we will follow the order of operations and also simplify where possible.
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Rewrite the division as multiplication by the reciprocal: \[ (-2) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(-\frac{5}{6}\right) \times \left(\frac{4}{3}\right) \]
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Now we can group all the multiplications together: \[ (-2) \times \left(-\frac{5}{6}\right) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(\frac{4}{3}\right) \]
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Let's simplify each multiplication step-by-step:
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First, calculate \((-2) \times \left(-\frac{5}{6}\right)\): \[ -2 \times -\frac{5}{6} = \frac{10}{6} = \frac{5}{3} \]
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Now, multiply by \(\frac{1}{5}\): \[ \frac{5}{3} \times \frac{1}{5} = \frac{5 \times 1}{3 \times 5} = \frac{1}{3} \]
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Now, multiply by \(\frac{3}{2}\): \[ \frac{1}{3} \times \frac{3}{2} = \frac{1 \times 3}{3 \times 2} = \frac{1}{2} \]
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Finally, multiply by \(\frac{4}{3}\): \[ \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3} \]
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So, the final answer is \(\frac{2}{3}\).