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What number should be subtracted from each of the numbers

    \[\mathbf{23},\text{ }\mathbf{30},\text{ }\mathbf{57}\text{ }\mathbf{and}\text{ }\mathbf{78}\]

so that the remainders are in proportion?
Class 10 | Exercise 1 | ICSE | Maths | ML Aggarwal | Ratio and Proportion
What number should be subtracted from each of the numbers

    \[\mathbf{23},\text{ }\mathbf{30},\text{ }\mathbf{57}\text{ }\mathbf{and}\text{ }\mathbf{78}\]

so that the remainders are in proportion?

Consider x be subtracted from each term

    \[23\text{ }\text{ }x,\text{ }30\text{ }\text{ }x,\text{ }57\text{ }\text{ }x\text{ }and\text{ }78\text{ }\text{ }x\]

are proportional

It can be written as

    \[\begin{array}{*{35}{l}} 23\text{ }\text{ }x:\text{ }30\text{ }\text{ }x\text{ }::\text{ }57\text{ }\text{ }x:\text{ }78\text{ }\text{ }x \\ \left( 23\text{ }\text{ }x \right)/\text{ }\left( 30\text{ }\text{ }x \right)\text{ }=\text{ }\left( 57\text{ }\text{ }x \right)/\text{ }\left( 78\text{ }\text{ }x \right) \\ \end{array}\]

By cross multiplication

    \[\left( 23\text{ }\text{ }x \right)\text{ }\left( 78\text{ }\text{ }x \right)\text{ }=\text{ }\left( 30\text{ }\text{ }x \right)\text{ }\left( 57\text{ }\text{ }x \right)\]

By further calculation

    \[\begin{array}{*{35}{l}} 1794\text{ }\text{ }23x\text{ }\text{ }78x\text{ }+\text{ }{{x}^{2}}~=\text{ }1710\text{ }\text{ }30x\text{ }\text{ }57x\text{ }+\text{ }{{x}^{2}} \\ {{x}^{2}}~\text{ }101x\text{ }+\text{ }1794\text{ }\text{ }{{x}^{2}}~+\text{ }87x\text{ }\text{ }1710\text{ }=\text{ }0 \\ \end{array}\]

So we get

    \[\begin{array}{*{35}{l}} \text{ }14x\text{ }+\text{ }84\text{ }=\text{ }0 \\ 14x\text{ }=\text{ }84 \\ x\text{ }=\text{ }84/14\text{ }=\text{ }6 \\ \end{array}\]

Therefore,

    \[6\]

is the number to be subtracted from each of the numbers.

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