Find the fraction which becomes $\dfrac{1}{2}$ when its numerator is increased by 6 and is equal to $\dfrac{1}{3}$ when its denominator is increased by 7.

Let the numerator be x and denominator be y.

Thus, fraction = $\dfrac{x}{y}$

Given,

The fraction becomes $\dfrac{1}{2}$ when its numerator is increased by 6.

⇒ $\dfrac{x + 6}{y} = \dfrac{1}{2}$

⇒ 2(x + 6) = y

⇒ 2x + 12 = y     ……….(1)

Given,

The fraction becomes $\dfrac{1}{3}$ when its denominator is increased by 7.

⇒ $\dfrac{x}{y + 7} = \dfrac{1}{3}$

⇒ 3x = y + 7     …….(2)

Substituting value of y from equation (1) in (2), we get :

⇒ 3x = (2x + 12) + 7

⇒ 3x = 2x + 19

⇒ 3x - 2x = 19

⇒ x = 19.

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

⇒ 2 × 19 + 12 = y

⇒ 38 + 12 = y

⇒ y = 50.

Hence, the fraction = $\dfrac{19}{50}$.