The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3,

Solution:

Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$

The sum of the numerator and denominator of the fraction is 12. Thus, we have

$x+y=12$

 

$\Rightarrow x+y-12=0$

If the denominator is increased by 3, the fraction becomes $\frac{1}{2}$. Thus, we have

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

$\Rightarrow 2 x=y+3$

 

$\Rightarrow 2 x-y-3=0$

So, we have two equations

$x+y-12=0$

 

$2 x-y-3=0$

Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

$\frac{x}{\mid \times(-3)-(-1) \times(-12)}=\frac{-y}{\mid \times(-3)-2 \times(-12)}=\frac{1}{\mid \times(-1)-2 \times 1}$

$\Rightarrow \frac{x}{-3-12}=\frac{-y}{-3+24}=\frac{1}{-1-2}$

$\Rightarrow \frac{x}{-15}=\frac{-y}{21}=\frac{1}{-3}$

$\Rightarrow \frac{x}{15}=\frac{y}{21}=\frac{1}{3}$

$\Rightarrow x=\frac{15}{3}, y=\frac{21}{3}$

 

$\Rightarrow x=5, y=7$

Hence, the fraction is $\frac{5}{7}$.