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 can follow the order of operations, as well as convert the division into multiplication by the reciprocal.
Let's break it down 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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Combine the entire expression: \[ = (-2) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(-\frac{5}{6}\right) \times \left(\frac{4}{3}\right) \]
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Now we can calculate it step by step:
First, multiply \( (-2) \) and \( -\frac{5}{6} \): \[ (-2) \times \left(-\frac{5}{6}\right) = \frac{10}{6} = \frac{5}{3} \]
Next, multiply with \( \frac{1}{5} \): \[ \frac{5}{3} \times \frac{1}{5} = \frac{5 \times 1}{3 \times 5} = \frac{1}{3} \]
Now multiply this by \( \frac{3}{2} \): \[ \frac{1}{3} \times \frac{3}{2} = \frac{1 \times 3}{3 \times 2} = \frac{3}{6} = \frac{1}{2} \]
Finally, multiply by \( \frac{4}{3} \): \[ \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3} \]
Thus, the final result of the expression is: \[ \boxed{\frac{2}{3}} \]