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 rearrange the expression as needed.
Let's break it down step-by-step:
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We can 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 the numbers 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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Now we can calculate step by step:
- First, calculate \((-2) \times \left(-\frac{5}{6}\right)\): \[ = \frac{10}{6} = \frac{5}{3} \]
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Next, we'll multiply that result by \(\left(\frac{1}{5}\right)\): \[ \frac{5}{3} \times \frac{1}{5} = \frac{5 \cdot 1}{3 \cdot 5} = \frac{1}{3} \]
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Now we multiply it by \(\left(\frac{3}{2}\right)\): \[ \frac{1}{3} \times \frac{3}{2} = \frac{1 \cdot 3}{3 \cdot 2} = \frac{3}{6} = \frac{1}{2} \]
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Finally, multiply that result by \(\left(\frac{4}{3}\right)\): \[ \frac{1}{2} \times \frac{4}{3} = \frac{1 \cdot 4}{2 \cdot 3} = \frac{4}{6} = \frac{2}{3} \]
Thus, the final result is: \[ \frac{2}{3} \]
So, the answer is \(\frac{2}{3}\).