(i)

  \((\frac{5}{9}\div\frac{1}{3})\) \(\div\frac{5}{2}= \frac{5}{9}\div(\frac{1}{3}\div\frac{5}{2})\)

LHS = \((\frac{5}{9}\div\frac{1}{3})\div\frac{5}{2}\)

= \((\frac{5}{9}\times\frac{3}{1}) \div\frac{5}{2}\)

= \((\frac{5\times3}{9\times1})\div\frac{5}{2}\)

= \(\frac{15}{9}\div\frac{5}{2}\)

= \(\frac{15}{9}\times\frac{2}{5}\)

= \(\frac{15\times2}{9\times5}\)

= \(\frac{30}{45}=\frac{30\div15}{45\div15}= \frac{2}{3}\)

RHS = \(\frac{5}{9}\div(\frac{1}{3}\div\frac{5}{2})\)

= \(\frac{5}{9}\div(\frac{1}{3}\times\frac{2}{5})\)

= \(\frac{5}{9}\div\frac{2}{15}\)

= \(\frac{5\times15}{9\times2}\)

= \(\frac{75}{18}= \frac{75\div3}{18\div3}=\frac{25}{6}\)

RHS ≠ LHS

Hence, False

(ii)

\(\{(-16)\div\frac{6}{5}\}\div\frac{-9}{10}= (-16)\div\{\frac{6}{5}\div\frac{-9}{10}\}\)

LHS = \((-16\div\frac{6}{5})\div\frac{-9}{10}\)

= \((\frac{-16}{1}\div\frac{6}{5})\div\frac{-9}{10}\)

= \((\frac{-16}{1}\times\frac{5}{6})\div\frac{-9}{10}\)

= \((\frac{-16\times5}{1\times6})\div\frac{-9}{10}\)

= \(\frac{-80}{6}\div\frac{-9}{10}\)

= \(\frac{-80}{6}\times\frac{-9}{10}\)

= \(\frac{-80\times10}{6\times-9}\)

= \(\frac{-800}{-54}=\frac{800}{54}= \frac{800\div2}{54\div2}=\frac{400}{27}\)

RHS = \(-16\div(\frac{6}{5}\div\frac{-9}{10})\)

= \(\frac{-16}{1}\div(\frac{6}{5}\times\frac{10}{-9})\)

= \(\frac{-16}{1}\div\frac{60}{-45}\)

= \(\frac{-16}{1}\times\frac{-45}{60}\)

= \(\frac{-16\times-45}{1\times60}\)

= \(\frac{720}{60}=\frac{720\div60}{60\div60}=12\)

RHS ≠ LHS 

Hence, False

(iii)

\((\frac{-3}{5}\div\frac{-12}{35})\div\frac{1}{14}= \frac{-3}{5}\div(\frac{-12}{35}\div\frac{1}{14})\)

LHS = \((\frac{-3}{5}\div\frac{-12}{35})\div\frac{1}{14}\)

= \((\frac{-3}{5}\times\frac{35}{-12})\div\frac{1}{14}\)

= \((\frac{-3\times35}{5\times-12}) \div\frac{1}{14}\)

= \(\frac{-105}{-60}\div\frac{1}{14}\)

= \(\frac{-105}{-60}\times\frac{14}{1}\)

= \(\frac{-105\times14}{-60\times1}\)

= \(\frac{-1470}{-60}=\frac{-1470}{-60}=\frac{1470\div30}{60\div30}\) = \(\frac{49}{2}\)

RHS = \(\frac{-3}{5}\div(\frac{-12}{35}\div\frac{1}{14})\)

= \(\frac{-3}{5}\div(\frac{-12}{35}\times\frac{14}{1})\)

= \(\frac{-3}{5}\div(\frac{-12\times1+14\times35}{35})\)

= \(\frac{-3}{5}\div\frac{-12+490}{35}\)

= \(\frac{-3}{5}\div\frac{478}{35}\)

= \(\frac{-3}{5}\times\frac{35}{478}\)

= \(\frac{-3\times35}{5\times478}\)

= \(\frac{-105}{2390}=\frac{-105\div5}{2390\div5}=\frac{-21}{478}\)

RHS ≠ LHS 

Hence, False