A fraction becomes $\dfrac{1}{2}$ when 1 is subtracted from its numerator and 1 is added to its denominator; it becomes $\dfrac{1}{3}$ when 6 is subtracted from its numerator and 1 from its denominator. Find the original fraction.

Let the numerator be x and denominator be y.

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

Given,

The fraction becomes $\dfrac{1}{2}$ when 1 is subtracted from the numerator and 1 is added to the denominator,

⇒ $\dfrac{x - 1}{y + 1} = \dfrac{1}{2}$

⇒ 2(x - 1) = y + 1

⇒ 2x - 2 = y + 1

⇒ y = 2x - 2 - 1

⇒ y = 2x - 3     ………(1)

Given,

The fraction becomes $\dfrac{1}{3}$ when 6 is subtracted from the numerator and 1 from the denominator,

⇒ $\dfrac{x - 6}{y - 1} = \dfrac{1}{3}$

⇒ 3(x - 6) = y - 1

⇒ 3x - 18 = y - 1

⇒ y = 3x - 18 + 1

⇒ y = 3x - 17     ………(2)

From equations (1) in (2), we get :

⇒ 3x - 17 = 2x - 3

⇒ 3x - 2x = -3 + 17

⇒ 3x - 2x = 14

⇒ x = 14.

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

⇒ y = 2(14) - 3 = 28 - 3 = 25.

Fraction = $\dfrac{x}{y} = \dfrac{14}{25}$.

Hence, the fraction = $\dfrac{14}{25}$.