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 and simplify step by step.
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Convert Division to Multiplication: Recall that dividing by a fraction is the same as multiplying by its reciprocal. \[ \div \left(\frac{2}{3}\right) \text{ becomes } \times \left(\frac{3}{2}\right) \] \[ \div \left(\frac{3}{4}\right) \text{ becomes } \times \left(\frac{4}{3}\right) \]
So the expression can be rewritten as: \[ (-2) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(-\frac{5}{6}\right) \times \left(\frac{4}{3}\right) \]
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Grouping and Simplifying: We can now group the multiplication: \[ (-2) \times \left(-\frac{5}{6}\right) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(\frac{4}{3}\right) \]
Start with the first two terms: \[ (-2) \times \left(-\frac{5}{6}\right) = \frac{10}{6} = \frac{5}{3} \]
Now, multiply this result by \(\frac{1}{5}\): \[ \frac{5}{3} \times \frac{1}{5} = \frac{5 \times 1}{3 \times 5} = \frac{1}{3} \]
Next, multiply 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 answer is \(\frac{2}{3}\).