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 handle the multiplication and division step by step.
- Convert divisions to multiplication by using the reciprocal: \[ a \div b = a \times \frac{1}{b} \]
So we rewrite the expression: \[ (-2) \times \left(\frac{1}{5}\right) \times \frac{3}{2} \times \left(- \frac{5}{6}\right) \times \frac{4}{3} \]
-
Combine everything together: We can group the terms that we are multiplying: \[ = (-2) \times \left(\frac{1}{5}\right) \times \left(- \frac{5}{6}\right) \times \left(\frac{3}{2}\right) \times \left(\frac{4}{3}\right) \]
-
Calculate the product: Let's multiply the numbers step by step.
Calculate the constants: \[ (-2) \times (-5) = 10 \] So we have: \[ 10 \times \frac{1}{5} \times \frac{3}{2} \times \frac{4}{3} \]
Multiplying \(\frac{1}{5}\) and \(10\): \[ 10 \times \frac{1}{5} = 2 \]
Now we rewrite the expression: \[ 2 \times \frac{3}{2} \times \frac{4}{3} \]
Now multiply \(2\) and \(\frac{3}{2}\): \[ 2 \times \frac{3}{2} = 3 \]
Now we have: \[ 3 \times \frac{4}{3} \]
Finally, multiply \(3\) and \(\frac{4}{3}\): \[ 3 \times \frac{4}{3} = 4 \]
So, the final result is: \[ \boxed{4} \]