Question

The numerator of a fraction is 6 less than its denominator. if 3 is added to the numerator the fraction becomes 2/3 . find the original fraction​

Answer 1

${\purple{\underline{\underline{\pink{\boxed{\boxed{\red{\star{\sf Answer:}}}}}}}}}$

${\blue{\underline{\orange{\star{\rm {The\:original\:fraction\:is = \dfrac{3}{9}}}}}}}$

${\sf{\large{\underline{\pink{Given:}}}}}$

➛The numerator of a fraction is 6 less than

its denominator.

➛ If 3 is added to the numerator the

fraction becomes 2/3 .

${\sf{\large{\underline{\pink{To \ Find :}}}}}$

The original fraction.

${\purple{\underline{\underline{\orange{\boxed{\boxed{\green{\star{\sf Solution:}}}}}}}}}$

Let the denominator = m

Hence, According to first condition,

Numerator = x - 6

$\implies \rm Original\:fraction\: = \dfrac{( m - 6 ) }{ m }$--( 1 )

According to the second condition,

$: \implies \rm \dfrac{( m - 6 + 3 ) }{m} = \dfrac{2}{3}$

$: \implies \rm \dfrac{( m - 3 ) }{m} = \dfrac{2}{3}$

$: \implies \rm {3( m - 3 ) } = 2m$

$: \implies \rm 3m - 9 = 2m$

$: \implies \rm 3m - 2m = 9$

${\large{\longrightarrow{ \green{ \boxed {\pink{ \star{ \rm {m = 9 }}}}}}}}$

Put m = 9 in equation ( 1 ) ,

$: \implies \rm Original\:fraction = \dfrac{ (x - 6 ) }{ x}$

$: \implies \rm \dfrac{( 9 - 6 )}{ 9}$

$: \implies \rm \dfrac{3}{ 9}$

${\large{\longrightarrow{\blue{\boxed{ \red{ \star{ \rm { Original\:fraction = \dfrac{3}{ 9}}}}}}}}}$

Answer 2

Let , The fraction be x/y

Given that ,

The numerator of a fraction is 6 less than its denominator

Thus ,

$\rm x = (y - 6) - - - (i)$

If 3 is added to the numerator the fraction becomes 2/3

Thus ,

$\rm \hookrightarrow \frac{(y - 6) +3 }{y} = \frac{2}{3} \\ \\ \rm \hookrightarrow 3y - 9 = 2y \\ \\ \rm \hookrightarrow y = 9$

Put the value of y = 9 in eq (i) , we get

$\rm \hookrightarrow x = 9 - 6 \\ \\ \rm \hookrightarrow </p><p>x = 3</p><p>$

$\therefore \sf \underline{The \: fraction \: is \: \frac{3}{9} \: or \: \frac{1}{3} }$