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(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 \[\left( \mathbf{15a}\text{ }\text{ }\mathbf{3b} \right):\text{ }\left( \mathbf{9a}\text{ }+\text{ }\mathbf{5b} \right)\].
Class 10 | Exercise 1 | ICSE | Maths | ML Aggarwal | Ratio and Proportion
(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, \[\left( 15a\text{ }\text{ }3b \right):\text{ }\left( 9a\text{ }+\text{ }5b \right)\text{ }=\text{ }6:\text{ }41\].

i

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