(i) If \[\mathbf{A}:\text{ }\mathbf{B}\text{ }=\text{ }\mathbf{1}/\mathbf{4}:\text{ }\mathbf{1}/\mathbf{5}\text{ }\mathbf{and}\text{ }\mathbf{B}:\text{ }\mathbf{C}\text{ }=\text{ }\mathbf{1}/\mathbf{7}:\text{ }\mathbf{1}/\mathbf{6}\], find A: B: C. (ii) If \[\mathbf{3A}\text{ }=\text{ }\mathbf{4B}\text{ }=\text{ }\mathbf{6C}\], find A: B: C

(i) We know that

\[\begin{array}{*{35}{l}}

A:\text{ }B\text{ }=\text{ }1/4\text{ }\times \text{ }5/1\text{ }=\text{ }5/4  \\

B:\text{ }C\text{ }=\text{ }1/7\text{ }\times \text{ }6/1\text{ }=\text{ }6/7  \\

\end{array}\]

Here the LCM of B terms \[4\text{ }and\text{ }6\text{ }is\text{ }12\]

Now making terms of B as \[12\]

\[\begin{array}{*{35}{l}}

A/B\text{ }=\text{ }\left( 5\text{ }\times \text{ }3 \right)/\text{ }\left( 4\text{ }\times \text{ }3 \right)\text{ }=\text{ }15/12\text{ }=\text{ }15:\text{ }12  \\

B/C\text{ }=\text{ }\left( 6\text{ }\times \text{ }2 \right)/\text{ }\left( 7\text{ }\times \text{ }2 \right)\text{ }=\text{ }12/14\text{ }=\text{ }12:\text{ }14  \\

\end{array}\]

So \[A:\text{ }B:\text{ }C\text{ }=\text{ }15:\text{ }12:\text{ }14\]

(ii) It is given that

\[3A\text{ }=\text{ }4B\]

We can write it as

\[\begin{array}{*{35}{l}}

A/B\text{ }=\text{ }4/3  \\

A:\text{ }B\text{ }=\text{ }4:\text{ }3  \\

\end{array}\]

Similarly \[4B\text{ }=\text{ }6C\]

We can write it as

\[\begin{array}{*{35}{l}}

B/C\text{ }=\text{ }6/4\text{ }=\text{ }3/2  \\

B:\text{ }C\text{ }=\text{ }3:\text{ }2  \\

\end{array}\]

So we get

\[A:\text{ }B:\text{ }C\text{ }=\text{ }4:\text{ }3:\text{ }2\]