Correct Answer: 0.007 and 0.008

Solution : Given: $\left(1 \frac{1}{3} \div 2 \frac{6}{7}\right.$ of $\left.5 \frac{3}{5}\right) \times\left(6 \frac{2}{5} \div 4 \frac{1}{2}\right.$ of $\left.5 \frac{1}{3}\right) \div\left(\frac{3}{4} \times 2 \frac{2}{3} \div \frac{5}{9}\right.$ of $\left.1 \frac{1}{5}\right)=k$
⇒ $\left(\frac{4}{3} \div \frac{20}{7}\right.$ of $\left.\frac{28}{5}\right) \times\left( \frac{32}{5} \div \frac{9}{2}\right.$ of $\left.\frac{16}{3}\right) \div\left(\frac{3}{4} \times \frac{8}{3} \div \frac{5}{9}\right.$ of $\left. \frac{6}{5}\right)=k$
⇒ $(\frac{4}{3} \div 16) \times( \frac{32}{5} \div 24) \div (\frac{3}{4} \times \frac{8}{3} \div \frac{2}{3})=k$
⇒ $\frac{1}{12} \times \frac{4}{15} \div (\frac{3}{4} \times 4)=k$
⇒ $\frac{1}{12} \times \frac{4}{15} \div 3=k$
⇒ $k = \frac{4}{540} = 0.0074$
⇒ $k$ lies between 0007 and 0.008.
Hence, the correct answer is 0.007 and 0.008.