Question

The numerator of a fraction is 5 less than its denominator. If 2 is subtracted from the numerator and 2 is

added to the denominator, the fraction becomes 2/5. find the original fraction.​

Answer 1

$\bf \underline{ \underline{\maltese\:Given} }$

$\sf\implies The \: numerator \: of \: a \: fraction \: is \: 5 \: less \: than \: its \: denominator.$

If 2 is subtracted from the numerator and 2 is added to the denominator.

$\sf\implies \bf \underline{Then}, \sf \: the \: fraction \: becomes \: \dfrac{2}{5}$

$\bf \underline{ \underline{\maltese\:To \: find } }$

$\sf \implies Original \: fraction = \: ?$

$\bf \underline{ \underline{\maltese\:Solution } }$

$\sf Let \: us \: assume \: that,$

$\sf \implies Numerator \: of \: the \: fraction \: be \: x.$

$\sf \implies Then, the \: denominator \: will \: be \: x + 5$

$\sf According \: to \: question,$

$\sf Equation = \bf \red{ \dfrac{x - 2}{(x + 5) + 2} = \dfrac{2}{5} }$

$\sf \implies \dfrac{x - 2}{x + 7} = \dfrac{2}{5}$

$\sf Doing \: cross \: multiplication,$

$\sf \implies (x - 2) \times 5 = (x + 7) \times 2$

$\sf \implies 5x - 10 = 2x + 14$

$\sf \implies 5x - 10 - 2x = 14$

$\sf \implies 3x - 10 = 14$

$\sf \implies 3x =24$

$\sf \implies x = \cancel\dfrac{24}{3}$

$\sf \implies x = 8$

$\bf \underline{Therefore},$

$\sf \implies Numerator \: (x) = \bf 8$

$\sf \implies Denominator\: (x + 5 ) = \bf (8 + 5) = 13$

$\underline{ \boxed{ \bf \red{Hence, \: the \: original \: fraction \: is \: \dfrac{8}{13} }}}$

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$\bf \underline{ \underline{\maltese\: Varification } }$

To verify the answer just write 8 in place of x in the equation.

$\sf \implies \dfrac{x - 2}{(x + 5) + 2} = \dfrac{2}{5}$

$\sf \implies \dfrac{8 - 2}{(8 + 5) + 2} = \dfrac{2}{5}$

$\sf \implies \dfrac{8 - 2}{13 + 2} = \dfrac{2}{5}$

$\sf \implies \cancel{ \dfrac{6}{15}} = \dfrac{2}{5}$

$\sf \implies \dfrac{2}{5} = \dfrac{2}{5}$

LHS and RHS are equal.

$\bold{\longrightarrow} \: \large{\tt \red{ Hence\: Verified \:!}}$