A girl goes to her friend's house, which is at a distance of 12 km. She covers half of the distance at a speed of x km/hr and the remaining distance at a speed of (x + 2) km/hr. If she takes 2 hrs 30 minutes to cover the whole distance, find 'x'.

Half of the distance i.e. 6 km is covered at a speed of x km/hr and remaining half at (x + 2) km/hr.

Time taken in each case = $\dfrac{6}{x}$ and $\dfrac{6}{x + 2}$

According to question,

$\Rightarrow \dfrac{6}{x} + \dfrac{6}{x + 2} = \dfrac{150}{60} \\[1em] \Rightarrow \dfrac{6(x + 2) + 6x}{x(x + 2)} = \dfrac{5}{2} \\[1em] \Rightarrow \dfrac{6x + 12 + 6x}{x^2 + 2x} = \dfrac{5}{2} \\[1em] \Rightarrow \dfrac{12x + 12}{x^2 + 2x} = \dfrac{5}{2} \\[1em] \Rightarrow 2(12x + 12) = 5(x^2 + 2x) \\[1em] \Rightarrow 24x + 24 = 5x^2 + 10x \\[1em] \Rightarrow 5x^2 + 10x - 24x - 24 = 0 \\[1em] \Rightarrow 5x^2 - 14x - 24 = 0 \\[1em] \Rightarrow 5x^2 - 20x + 6x - 24 = 0 \\[1em] \Rightarrow 5x(x - 4) + 6(x - 4) = 0 \\[1em] \Rightarrow (5x + 6)(x - 4) = 0 \\[1em] \Rightarrow 5x + 6 = 0\text{ or } x - 4 = 0 \\[1em] \Rightarrow x = -\dfrac{6}{5} \text{ or } x = 4.$

Since speed cannot be negative,

∴ x = 4.

Hence, the value of x = 4.