Let the numerator be A and the denominator be B.

So, the required fraction is $A/B$.

ATQ,

The sum of the numerator and denominator of the fraction is $12$.

$A+B=12$

⇒ $A+B–12=0$

ATQ,

If $3$ is added in the denominator, the fraction becomes $12$

Use this equation which is given in the question, 

$A/(B+3)=1/2$

⇒ $2A=(B+3)$

⇒ $2A-B–3=0$

⇒ $2A–B=3$

Hence, two equations are,

$A+B–12=0$…… (a)

$2A–B-3=0$…….. (b)

Add (a) and (b), we will get

$A+B–12+(2A–B–3)=0$

⇒ $3A-15=0$

⇒ $3A=15$

⇒ $A=5$

Use the obtained value of $A=5$ in (a), and find B

$5+B–12=0$

$5+B=12$

⇒ $B=7$

Therefore, the fraction is $5/7$.