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 simplify step by step.
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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 group and multiply together: \[ (-2) \times (-\frac{5}{6}) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(\frac{4}{3}\right) \]
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First, compute the multiplication of the first two terms: \[ (-2) \times (-\frac{5}{6}) = \frac{10}{6} = \frac{5}{3} \]
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Now multiply this result with \(\left(\frac{1}{5}\right)\): \[ \frac{5}{3} \times \frac{1}{5} = \frac{5 \times 1}{3 \times 5} = \frac{1}{3} \]
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Multiply this with \(\left(\frac{3}{2}\right)\): \[ \frac{1}{3} \times \frac{3}{2} = \frac{1 \times 3}{3 \times 2} = \frac{3}{6} = \frac{1}{2} \]
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Finally, multiply this result by \(\left(\frac{4}{3}\right)\): \[ \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3} \]
Thus, the final answer is: \[ \boxed{\frac{2}{3}} \]