The sum of a numerator and denominator of a fraction is 18.

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 the denominator of the fraction is 18 . Thus, we have

$x+y=18$

$\Rightarrow x+y-18=0$

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

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

$\Rightarrow 3 x=y+2$

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

So, we have two equations

$x+y-18=0$

$3 x-y-2=0$

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

By using cross-multiplication, we have

$\frac{x}{\mid \times(-2)-(-1) \times(-18)}=\frac{-y}{\mid \times(-2)-3 \times(-18)}=\frac{1}{\mid \times(-1)-3 \times 1}$

$\Rightarrow \frac{x}{-2-18}=\frac{-y}{-2+54}=\frac{1}{-1-3}$

$\Rightarrow \frac{x}{-20}=\frac{-y}{52}=\frac{1}{-4}$

$\Rightarrow \frac{x}{20}=\frac{y}{52}=\frac{1}{4}$

$\Rightarrow x=\frac{20}{4}, y=\frac{52}{4}$

$\Rightarrow x=5, y=13$

Hence, the fraction is $\frac{5}{13}$.