Question

$solve \: the \: problem \: please \: \\ write \: in \: paper \: nd \: send$
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Answer 1

$\Large{\bold{\blue{\underline{\red{A}\pink{ns}\green{we}\purple{r:-}}}}}$

$\rm{\mapsto The \: value \: of \: \dfrac{(p + q)}{(q - p)} \: is \: \dfrac{61}{38}}$

$\Large{\bold{\pink{\underline{\green{G}\purple{iv}\orange{en}\red{:-}}}}}$

$\rm{\leadsto 0.232323....}$

$\Large{\bold{\blue{\underline{\red{To}\:\pink{Fin}\green{d}\purple{:-}}}}}$

$\rm{\leadsto The \: value \: of \: \dfrac{(p + q)}{(q - p)} = \: ?}$

$\Large{\bold{\pink{\underline{\red{So}\purple{lut}\green{ion}\orange{:-}}}}}$

$\tt{:\implies x = \dfrac{p}{q}}$

$\tt{:\implies \dfrac{p}{q} = 0.232323....}$

$\tt{:\implies \dfrac{p}{q} = 0. \overline{23}}$

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$\mathscr{L}et, \: \sf{x = 0.232323....}$

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$\underline{\bm{ \star \: According \: to \: the \: given \: question}}$

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$\tt{:\implies 100x = 23.232323....}$

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$\tt{:\implies 100x = 23 + 0.232323....}$

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$\tt{:\implies 100x = 23 + x}$

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$\tt{:\implies 100x - x = 23}$

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$\tt{:\implies 99x = 23}$

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$\bf{:\implies x = \dfrac{23}{99} }$

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$\tt{:\implies \dfrac{p}{q} = \dfrac{23}{99} }$

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$\qquad \: \: \rm{\tiny{ \dag By \: componendo \: divindo \: rule}}$

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$\tt{:\implies \dfrac{p + q}{p - q} = \dfrac{23 + 99}{23 - 99} }$

⠀⠀

$\tt{:\implies \dfrac{p + q}{p - q} = \dfrac{122}{ - 76} }$

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$\tt{:\implies \dfrac{p + q}{p - q} = \dfrac{ - 61}{38} }$

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$\tt{:\implies \cancel{-}\dfrac{p + q}{p - q} = \cancel{- } \dfrac{61}{38} }$

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$\tt{:\implies \red{\boxed{ \blue{ \bf{\dfrac{p + q}{q - p} = \dfrac{61}{38} }}}}}$