The denominator of a fraction is 3 more than the numerator. If 2 is added to the numerator and 5 is added to the denominator, the fraction becomes $\dfrac{1}{2}$. Find the fraction.

Let the numerator of the required fraction be x.

Then, its denominator = (x + 3).

The fraction is $\dfrac{x}{x + 3}$.

According to the question, if 2 is added to the numerator and 5 is added to the denominator, the fraction becomes $\dfrac{1}{2}$.

$\therefore \dfrac{x + 2}{(x + 3) + 5} = \dfrac{1}{2} \\[1em] \Rightarrow \dfrac{x + 2}{x + 8} = \dfrac{1}{2} \\[1em] \Rightarrow 2(x + 2) = 1(x + 8) \quad \text{[By cross-multiplication]} \\[1em] \Rightarrow 2x + 4 = x + 8 \\[1em] \Rightarrow 2x - x = 8 - 4 \quad \text{[Transposing +4 to RHS and +x to LHS]} \\[1em] \Rightarrow x = 4$

Numerator = x = 4

Denominator = (x + 3) = 4 + 3 = 7.

Hence, the required fraction is $\dfrac{4}{7}$.