(i) If \[\mathbf{3x}\text{ }+\text{ }\mathbf{5y}/\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{5y}\text{ }=\text{ }\mathbf{7}/\mathbf{3}\],find x: y. (ii) If \[\mathbf{a}:\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{3}:\text{ }\mathbf{11}\], find \[\left( \mathbf{15a}\text{ }\text{ }\mathbf{3b} \right):\text{ }\left( \mathbf{9a}\text{ }+\text{ }\mathbf{5b} \right)\].

(i) If

$\mathbf{3x}\text{ }+\text{ }\mathbf{5y}/\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{5y}\text{ }=\text{ }\mathbf{7}/\mathbf{3}$

,find x: y. (ii) If

$\mathbf{a}:\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{3}:\text{ }\mathbf{11}$

, find

$\left( \mathbf{15a}\text{ }\text{ }\mathbf{3b} \right):\text{ }\left( \mathbf{9a}\text{ }+\text{ }\mathbf{5b} \right)$

(i) If

$\mathbf{3x}\text{ }+\text{ }\mathbf{5y}/\text{ }\mathbf{3x}\text{ }\text{ }\mathbf{5y}\text{ }=\text{ }\mathbf{7}/\mathbf{3}$

,find x: y. (ii) If

$\mathbf{a}:\text{ }\mathbf{b}\text{ }=\text{ }\mathbf{3}:\text{ }\mathbf{11}$

, find

$\left( \mathbf{15a}\text{ }\text{ }\mathbf{3b} \right):\text{ }\left( \mathbf{9a}\text{ }+\text{ }\mathbf{5b} \right)$

(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,

$\left( 15a\text{ }\text{ }3b \right):\text{ }\left( 9a\text{ }+\text{ }5b \right)\text{ }=\text{ }6:\text{ }41$

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