The sum of the numerator and denominator of a fraction is 4 more than twice the numerator. If the numerator and denominator are increased by 3, they are in the ratio 2:3. Determine the fraction.

Suppose numerator = x

Also, denominator= y
Thus, the Fraction become,
$=\frac{\mathrm{x}}{\mathrm{y}}\mathrm{ }$From first condition of question,
$\Rightarrow \mathrm{x}+\mathrm{y}=2\mathrm{x}+4$
$\Rightarrow \mathrm{y}=\mathrm{x}+4 .......\left(1\right)$
From second condition of question,
$\frac{\mathrm{x}+3}{\mathrm{y}+3}=\frac{2}{3}$
$3\mathrm{x}+9=2\mathrm{y}+6$
$3\mathrm{x}-2\mathrm{y}=-3 .......\left(2\right)$
Put this value of y in equation (2) from equation (1)
$3\mathrm{x}-2\left(\mathrm{x}+4\right)=-3$
$3\mathrm{x}-2\mathrm{x}-8=-3$
$\mathrm{x}=5$
Put this value in equation (1),
$\Rightarrow \mathrm{y}=5+4$
$\Rightarrow \mathrm{y}=9$
So, the fraction become,
$\frac{\mathbf{x}}{\mathbf{y}}\mathbf{=}\frac{\mathbf{5}}{\mathbf{9}}$