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 step by step.
- Convert the division by a fraction into multiplication by its reciprocal:
\[ = (-2) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(-\frac{5}{6}\right) \times \left(\frac{4}{3}\right) \]
- Now, we can multiply all these factors together:
\[ = (-2) \times \left(\frac{1}{5}\right) \times \left(\frac{3}{2}\right) \times \left(-\frac{5}{6}\right) \times \left(\frac{4}{3}\right) \]
- Combine the numerical coefficients and fractions:
\[ = \frac{(-2) \times 1 \times 3 \times (-5) \times 4}{5 \times 2 \times 6 \times 3} \]
- Calculate the numerator:
\[ = (-2) \times (-5) \times 4 \times 3 = 2 \times 5 \times 4 \times 3 = 120 \]
- Calculate the denominator:
\[ = 5 \times 2 \times 6 \times 3 = 180 \]
- So, we have:
\[ = \frac{120}{180} \]
- Simplify the fraction:
\[ = \frac{2}{3} \]
Thus, the final answer is \(\frac{2}{3}\).
From the options provided, the correct response is:
Start Fraction 2 over 3 End Fraction.