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The sum of the numerator and the 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.
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Let the fraction be $\frac{x}{y}, y \neq 0 \therefore x + y = 4 + 2x \implies y = x + 4$ ----- (i) And $\frac{x+3}{y+3} = \frac{2}{3} \implies 3x - 2y + 3 = 0$ ------- (ii) Solving (i) and (ii), we get $x = 5, y = 9 \therefore$ Fraction is $\frac{5}{9}$