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, performing multiplications and divisions from left to right.
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First, calculate \((-2) \times \left(\frac{1}{5}\right)\): \[ -2 \times \frac{1}{5} = -\frac{2}{5}. \]
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Next, divide \(-\frac{2}{5} \div \left(\frac{2}{3}\right)\): Dividing by a fraction is the same as multiplying by its reciprocal: \[ -\frac{2}{5} \div \frac{2}{3} = -\frac{2}{5} \times \frac{3}{2} = -\frac{2 \times 3}{5 \times 2} = -\frac{6}{10} = -\frac{3}{5}. \]
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Then, multiply \(-\frac{3}{5} \times \left(-\frac{5}{6}\right)\): Since multiplying two negatives results in a positive: \[ -\frac{3}{5} \times -\frac{5}{6} = \frac{3 \times 5}{5 \times 6} = \frac{15}{30} = \frac{1}{2}. \]
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Finally, divide \(\frac{1}{2} \div \left(\frac{3}{4}\right)\): Again, dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3}. \]
Thus, the final answer is
\[ \frac{2}{3}. \]
So the correct response is:
\[ \frac{2}{3}. \]