(i) \[\left( x\text{ }\text{ }9 \right)/\text{ }\left( 3x\text{ }+\text{ }6 \right)\text{ }=\text{ }{{\left( 4/9 \right)}^{2}}\]
So we get
\[\left( x\text{ }\text{ }9 \right)/\text{ }\left( 3x\text{ }+\text{ }6 \right)\text{ }=\text{ }16/81\]
By cross multiplication
\[\begin{array}{*{35}{l}}
81x\text{ }\text{ }729\text{ }=\text{ }48x\text{ }+\text{ }96 \\
81x\text{ }\text{ }48x\text{ }=\text{ }96\text{ }+\text{ }729 \\
\end{array}\]
So we get
\[\begin{array}{*{35}{l}}
33x\text{ }=\text{ }825 \\
x\text{ }=\text{ }825/33\text{ }=\text{ }25 \\
\end{array}\]
(ii) \[\left( 3x\text{ }+\text{ }1 \right)/\text{ }\left( 5x\text{ }+\text{ }3 \right)\text{ }=\text{ }{{3}^{3}}/\text{ }{{4}^{3}}\]
So we get
\[\left( 3x\text{ }+\text{ }1 \right)/\text{ }\left( 5x\text{ }+\text{ }3 \right)\text{ }=\text{ }27/64\]
By cross multiplication
\[\begin{array}{*{35}{l}}
64\text{ }\left( 3x\text{ }+\text{ }1 \right)\text{ }=\text{ }27\text{ }\left( 5x\text{ }+\text{ }3 \right) \\
192x\text{ }+\text{ }64\text{ }=\text{ }135x\text{ }+\text{ }81 \\
192x\text{ }\text{ }135x\text{ }=\text{ }81\text{ }\text{ }64 \\
57x\text{ }=\text{ }17 \\
\end{array}\]
So we get
\[x\text{ }=\text{ }17/57\]