If the numerator of a fraction is multiplied by 2 and its denominator is increased by 1, it becomes 1. However, if the numerator is increased by 4 and denominator is multiplied by 2, the fraction becomes $\dfrac{1}{2}$. Find the fraction.

Let numerator be x and denominator be y.

Given,

If the numerator of a fraction is multiplied by 2 and its denominator is increased by 1, it becomes 1.

$\therefore \dfrac{2 \times x}{y + 1} = 1 \\[1em] \Rightarrow 2x = 1(y + 1) \\[1em] \Rightarrow 2x = y + 1 \\[1em] \Rightarrow 2x - y - 1 = 0 ………(1)$

Given,

If the numerator is increased by 4 and denominator is multiplied by 2, the fraction becomes $\dfrac{1}{2}$.

$\therefore \dfrac{x + 4}{2 \times y} = \dfrac{1}{2} \\[1em] \Rightarrow 2(x + 4) = 2y \times 1 \\[1em] \Rightarrow 2x + 8 = 2y \\[1em] \Rightarrow 2x - 2y + 8 = 0 ……..(2)$

Subtracting equation (2) from (1), we get :

⇒ 2x - y - 1 - (2x - 2y + 8) = 0

⇒ 2x - 2x - y + 2y - 1 - 8 = 0

⇒ y - 9 = 0

⇒ y = 9.

Substituting value of y in equation (1), we get :

⇒ 2x - 9 - 1 = 0

⇒ 2x - 10 = 0

⇒ 2x = 10

⇒ x = $\dfrac{10}{2}$ = 5.

Fraction = $\dfrac{x}{y} = \dfrac{5}{9}$.

Hence, fraction = $\dfrac{5}{9}$.