Question

The numerator of a fraction is 9 less than 2 times of its denominator. If 5 and 8 are added to the numerator and denominator respectively, the new fraction is equivalent to 2/ 3 . Find the original fraction

Answer 1

Solution :-

Let :-

Numerator = x

Denominator = y

According to the first condition :-

$\longrightarrow \sf x = 2y-9 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .....................(1)$

According to the second condition :-

$\longrightarrow \sf \dfrac{x+5}{y+8} = \dfrac{2}{3} \\\\\longrightarrow 3(x+5) = 2 (y+8) \\\\\longrightarrow 3x+15 = 2y + 16 \\\\\longrightarrow 3x-2y = 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .....................(2)$

Substitute the value of x in eq (2) :-

$\longrightarrow \sf 3 (2y-9)- 2y = 1 \\\\\longrightarrow 6y-27 - 2y = 1 \\\\\longrightarrow 6y-2y = 28 \\\\\longrightarrow 4y = 28 \\\\\longrightarrow \bf y = 7$

Substitute y = 7 in eq (1) :-

$\longrightarrow \sf x = 2 \times 7 - 9 \\\\\longrightarrow x = 14 - 9 \\\\\longrightarrow \bf x = 5$

$\bf FRACTION = \dfrac{5}{7}$

Answer 2

${ \bf{ \underline { \underline{Given \: data}}}:-}$

๏ The numerator of a fraction is 9 less than 2 times of its denominator.

๏ 5 and 8 are added to the numerator and denominator.

๏ The new fraction is equivalent to 2/3.

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

To find the original fraction :

Let, numerator of a fraction be x and denominator of a fraction be y.

{According to given}

→ Numerator = x = 2y - 9 .....( 1 )

→ Denominator = y .....( 2 )

Now, {According to given} 5 and 8 are added to the numerator and denominator.

{From ( 1 )}

→ Numerator = 2y - 9 + 5

→ Numerator = 2y - 4 .....( 3 )

{From ( 3 )}

→ Denominator = y + 8 .....( 4 )

Now, {According to given} the new fraction is equivalent to 2/3.

${ \dashrightarrow{ \bf{ \frac{Numerator}{Denominator} = \frac{2}{3} }}}$

Now {from ( 3 ) & ( 4 )}

${ \dashrightarrow{ \bf{ \frac{2y - 4}{y + 8} = \frac{2}{3} }}}$

${ \dashrightarrow{ \bf{ 3({2y - 4}) = 2({y + 8}) }}}$

${ \dashrightarrow{ \bf{ {6y - 12} = {2y + 16} }}}$

${ \dashrightarrow{ \bf{ {6y - 2y} = { 16 + 12} }}}$

${ \dashrightarrow{ \bf{ 4y = 28 }}}$

${ \dashrightarrow{ \bf{ y = \frac{28}{4} }}}$

${ \dashrightarrow{ \bf{ y = 7 }}}$

Put value of y in eq. ( 1 )

→ Numerator = x = 2y - 9

→ Numerator = x = 2( 7 ) - 9

→ Numerator = x = 14 - 9

→ Numerator = x = 5

Hence, the original fraction is

${ \dashrightarrow{ \bf{ \frac{Numerator}{Denominator} = \frac{5}{7} }}}$