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 and simplify it step by step.
Step 1: Simplify the expression by addressing multiplication and division.
We can start by rewriting the division as multiplication by the reciprocal:
\[ (-2) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(-\frac{5}{6}\right) \times \left(\frac{4}{3}\right) \]
Step 2: Group the multiplication and multipliers.
Now we'll combine the negative numbers and the fractions together:
- Multiply \((-2)\) and \(-\frac{5}{6}\) and then the fractions:
- \((-2) \times -\frac{5}{6} = \frac{10}{6} = \frac{5}{3}\)
- Now combine everything:
- \(\frac{5}{3} \times \frac{1}{5} \times \frac{3}{2} \times \frac{4}{3}\)
Step 3: Simplify the multiplication of the fractions.
Calculating this step-by-step:
\[ \left(\frac{5}{3} \times \frac{1}{5}\right) = \frac{5 \times 1}{3 \times 5} = \frac{1}{3} \]
Now we need to multiply that result with \(\frac{3}{2}\):
\[ \frac{1}{3} \times \frac{3}{2} = \frac{1 \times 3}{3 \times 2} = \frac{3}{6} = \frac{1}{2} \]
Now multiply this result with \(\frac{4}{3}\):
\[ \frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3} \]
Final Result
Thus, the final result of \((-2) \times \left(\frac{1}{5}\right) \div \left(\frac{2}{3}\right) \times \left(-\frac{5}{6}\right) \div \left(\frac{3}{4}\right)\) is
\[ \boxed{\frac{2}{3}} \]