(i) \[\mathbf{3x}\text{ }+\text{ }\mathbf{5y}/\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{5y}\text{ }=\text{ }\mathbf{7}/\mathbf{3}\]
By cross multiplication
\[9x\text{ }+\text{ }15y\text{ }=\text{ }21x\text{ }\text{ }35y\]
By further simplification
\[\begin{array}{*{35}{l}}
21x\text{ }\text{ }9x\text{ }=\text{ }15y\text{ }+\text{ }35y \\
12x\text{ }=\text{ }50y \\
\end{array}\]
So we get
\[x/y\text{ }=\text{ }50/12\text{ }=\text{ }25/6\]
Therefore, \[x:\text{ }y\text{ }=\text{ }25:\text{ }6\]
(ii) It is given that
\[\begin{array}{*{35}{l}}
a:\text{ }b\text{ }=\text{ }3:\text{ }11 \\
a/b\text{ }=\text{ }3/11 \\
\end{array}\]
It is given that
\[\left( \mathbf{15a}\text{ }\text{ }\mathbf{3b} \right):\text{ }\left( \mathbf{9a}\text{ }+\text{ }\mathbf{5b} \right)\]
Now dividing both numerator and denominator by b
\[=\text{ }\left[ 15a/b\text{ }\text{ }3b/b \right]/\text{ }\left[ 9a/b\text{ }+\text{ }5b/b \right]\]
By further calculation
\[=\text{ }\left[ 15a/b\text{ }\text{ }3 \right]/\text{ }\left[ 9a/b\text{ }+\text{ }5 \right]\]
Substituting the value of a/ b
\[=\text{ }\left[ 15\text{ }\times \text{ }3/11\text{ }\text{ }3 \right]/\text{ }\left[ 9\text{ }\times \text{ }3/11\text{ }+\text{ }5 \right]\]
So we get
\[=\text{ }\left[ 45/11\text{ }\text{ }3 \right]/\text{ }\left[ 27/11\text{ }+\text{ }5 \right]\]
Taking LCM
\[\begin{array}{*{35}{l}}
=\text{ }\left[ \left( 45\text{ }\text{ }33 \right)/\text{ }11 \right]/\text{ }\left[ \left( 27\text{ }+\text{ }55 \right)/\text{ }11 \right] \\
=\text{ }12/11/\text{ }82/11 \\
\end{array}\]
We can write it as
\[\begin{array}{*{35}{l}}
=\text{ }12/11\text{ }\times \text{ }11/82 \\
=\text{ }12/82 \\
=\text{ }6/41 \\
\end{array}\]
Hence, \[\left( 15a\text{ }\text{ }3b \right):\text{ }\left( 9a\text{ }+\text{ }5b \right)\text{ }=\text{ }6:\text{ }41\].