Question

.The sum of the numerator and the denominator of a given fraction is 12. If 3 is added to its denominator,
then the fraction becomes 1/2. Find the given fraction.
​

Answer 1

AnswEr :

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

The sum of the numerator and the denominator of a given fraction is 12. If 3 is added to its denominator, then the fraction becomes 1/2.

$\bf{\large{\underline{\sf{To\:find\::}}}}$

The fraction.

$\bf{\Large{\underline{\tt{\orange{Explanation\::}}}}}$

Let the numerator be r

Let the denominator be m

Fraction = $\bf\dfrac{r}{m} }$

$\bf{\Large{{\underline{\underline{\tt{\red{A.T.Q\::}}}}}}}$

$\mapsto\tt{r+m=12}\\\\\\\mapsto\tt{\orange{r\:=\:12-m.....................(1)}}$

&

$\mapsto\tt{\dfrac{r}{m+3} =\dfrac{1}{2}}\\ \\\\\\\\\mapsto\tt{2r=m+3}\\\\\\\\\mapsto\tt{2(12-m)=m+3\:\:\:\:\:\:\:\:\big[from(1)\big]}\\\\\\\\\mapsto\tt{24-2m=m+3}\\\\\\\\\mapsto\tt{-2m-m=3-24}\\\\\\\\\mapsto\tt{-3m=-21}\\\\\\\\\mapsto\tt{m\:=\:\cancel{\dfrac{-21}{-3} }}\\\\\\\\\mapsto\tt{\red{m\:=\:7}}$

Now,

from equation (1), we get;

$\mapsto\tt{r\:=\:12-7}\\\\\\\mapsto\tt{\red{r\:=\:5}}$

$\therefore \bf{The\:fraction\:is\:=\dfrac{r}{m} =\dfrac{5}{7} }$

Answer 2

To find: We have to find the fraction in the given problem.

Given: Sum of the numerator and denominator of a fraction is 12. If 3 is added to its denominator, the fraction becomes 1/2.

Solution: Let The numerator be x and Denominator be y.

Fraction = $\frac{x}{y}$

From the given condition,

x + y = 12

X = 12 - y ------- (i)

-------------------------

If the denominator is increased by 3, then the fraction becomes 1/2.

$\frac{x}{y + 3} = \frac{1}{2 }$

$2x = y + 3$

2x = y + 3 ---- (ii)

--------------------------

Now,

Adding values of x from equation (i) in (ii).

$2(12 - y) = y + 3$

$24 - 2y = y + 3$

$3y = 21$

$y = 7$

Putting y = 7 in equation (ii).

$x = 12 - 7 \\ x = 5$

Therefore, X = 5 And Y = 7.

_____________________________

Fraction = $\frac{x}{y} = \frac{5}{7}$