Find the fraction which becomes $\dfrac{1}{2}$ when the denominator is increased by 4 and is equal to $\dfrac{1}{8}$ when the numerator is diminished by 5.

Let numerator be x and denominator be y.

Given, fraction becomes $\dfrac{1}{2}$ when the denominator is increased by 4.

$\therefore \dfrac{x}{y + 4} = \dfrac{1}{2}$

⇒ 2x = y + 4

⇒ 2x - y = 4 …….(i)

Given, fraction becomes $\dfrac{1}{8}$ when the numerator is diminished by 5

$\therefore \dfrac{x - 5}{y} = \dfrac{1}{8}$

⇒ 8(x - 5) = y ……(ii)

Substituting value of y from eq. (ii) in (i) we get,

⇒ 2x - [8(x - 5)] = 4

⇒ 2x - 8x + 40 = 4

⇒ -6x = 4 - 40

⇒ 6x = 36

⇒ x = 6.

Substituting value of x in (ii) we get,

⇒ 8(6 - 5) = y

⇒ y = 8(1) = 8.

Original fraction = $\dfrac{x}{y} = \dfrac{6}{8}$

Hence, fraction = $\dfrac{6}{8}$.