To solve the expression \((-3) \div \left(\frac{3}{4}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \div \left(\frac{5}{7}\right)\), we will follow the order of operations and simplify step by step.
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Convert division by a fraction into multiplication by its reciprocal. Thus, we can express the division as: \[ (-3) \div \left(\frac{3}{4}\right) = (-3) \times \left(\frac{4}{3}\right) \]
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Substitute that back into the expression: \[ (-3) \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \div \left(\frac{5}{7}\right) \] which can be rewritten as: \[ (-3) \times \left(\frac{4}{3}\right) \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \times \left(\frac{7}{5}\right) \]
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Next, we simplify the entire expression:
- We will first deal with \((-3) \times \left(\frac{4}{3}\right)\): \[ (-3) \times \left(\frac{4}{3}\right) = -\frac{12}{3} = -4 \]
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Now we have: \[ -4 \times \left(\frac{2}{7}\right) \times \left(-\frac{5}{16}\right) \times \left(\frac{7}{5}\right) \]
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Multiply \(-4\) and \(-\frac{5}{16}\) together: \[ -4 \times -\frac{5}{16} = \frac{20}{16} = \frac{5}{4} \]
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Now the expression simplifies to: \[ \frac{5}{4} \times \left(\frac{2}{7}\right) \times \left(\frac{7}{5}\right) \]
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Next, multiply the fractions: \[ \frac{5}{4} \times \frac{2}{7} = \frac{10}{28} = \frac{5}{14} \]
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Now multiply by \(\frac{7}{5}\): \[ \frac{5}{14} \times \frac{7}{5} = \frac{35}{70} = \frac{1}{2} \]
Thus, the final answer is: \[ \boxed{\frac{1}{2}} \]