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 break it down step by step, using the properties of operations and rules for multiplying and dividing.
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Reorganize the expression: We need to understand that division by a fraction is the same as multiplying by its reciprocal.
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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Combine the fractions: Now multiply all factors together in the same order: \[ (-2) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(-\frac{5}{6}\right) \times \left(\frac{4}{3}\right) \]
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Group the factors: Let's group the constants and simplify step by step: \[ (-2) \times (-\frac{5}{6}) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(\frac{4}{3}\right) \]
We can re-arrange the multiplication: \[ (-2) \times (-5) \times (4) \div (5) \div (6) \div (2) \div (3) \]
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Perform the calculation:
- First, compute the multiplication of the negatives and remaining numbers: \[ (-2) \times (-5) = 10 \]
So we have: \[ 10 \times 4 = 40 \]
Now we have: \[ \frac{40}{5 \times 6 \times 2 \times 3} \]
- Calculate the denominator: \[ 5 \times 6 = 30 \] \[ 30 \times 2 = 60 \] \[ 60 \times 3 = 180 \]
Therefore: \[ 40 \div 180 \]
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Simplify the fraction: \[ \frac{40}{180} \]
We can simplify this:
- The GCD of \(40\) and \(180\) is \(20\): \[ \frac{40 \div 20}{180 \div 20} = \frac{2}{9} \]
Thus, the final answer is: \[ \boxed{\frac{2}{9}} \]
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