$${\textbf{Step -1: Let numerator}}$$$$\mathbf{ = x,}$$ $${\textbf{denominator}}$$ $$= \mathbf {y,}$$ $${\textbf{fraction}}$$$$\mathbf{ = \dfrac{x}{y}.}$$
$${\text{Given, sum of numerator and denominator is 3 more than twice of the numerator}}{\text{.}}$$
$$ \Rightarrow x + y = 2x + 3$$
$$ \Rightarrow x - y = - 3 \ldots \left( 1 \right)$$
$${\textbf{Step -2: It is given that, if 4 is added to both numerator and denominator then their ratio becomes}}$$ $$\mathbf{3:4.}$$
$$ \Rightarrow \dfrac{{x + 4}}{{y + 4}} = \dfrac{3}{4}$$
$$ \Rightarrow 4\left( {x + 4} \right) = 3\left( {y + 4} \right)$$
$$ \Rightarrow 4x + 16 = 3y + 12$$
$$ \Rightarrow 4x - 3y = -4 \ldots \left( 2 \right)$$
$${\textbf{Step -3: Multiply equation }}\left(\mathbf1 \right){\textbf{ by - 4 and add it with equation }}\left( \mathbf2 \right)\textbf.$$
$$ \Rightarrow - 4x + 4y + 4x - 3y = 12 - 4$$
$$ \Rightarrow y = 8$$ $${\textbf{[Substitute it in equation }}\left( 1 \right)].$$
$$ \Rightarrow x - 8 = - 3$$
$$ \Rightarrow x = 5$$
$${\text{Therefore, }}$$
$${\text{Numerator x = 5}}$$
$${\text{Denominator y = 8}}$$
$$ \Rightarrow {\text{Fraction }}\dfrac{x}{y} = \dfrac{{5}}{{8}}$$
$${\textbf{Hence, the required fraction is }}\mathbf {\dfrac{{5}}{{8}}.}$$