Ashish goes to his friends house which is 12 km away from his house. He covers half of the distance at a speed of x km per hour and the remaining at (x + 2) km per hour. If he takes 2 hrs 30 min. to cover the whole distance, find the value of x.

Given,

Total distance = 12 km

In first case :

The speed for the first half = x km/hr.

Time = $\dfrac{\text{Distance}}{\text{Speed}}$

Time₁ = $\dfrac{6}{x}$

In second case :

Speed for second half = (x + 2) km/hr

Time₂ = $\dfrac{6}{x + 2}$

Time taken to cover the whole distance = 2 hrs 30 min.

$\Rightarrow 2.5 = \dfrac{6}{x} + \dfrac{6}{x + 2} \\[1em] \Rightarrow 2.5 = \dfrac{6(x + 2) + 6x}{x(x + 2)} \\[1em] \Rightarrow 2.5 = \dfrac{6x + 12 + 6x}{x^2 + 2x} \\[1em] \Rightarrow 2.5(x^2 + 2x) = 12x + 12 \\[1em] \Rightarrow 2.5x^2 + 5x = 12x + 12 \\[1em] \Rightarrow 2.5x^2 - 12x - 12 + 5x = 0 \\[1em] \Rightarrow 2.5x^2 - 7x - 12 = 0$

Multiplying by 2 on both sides of equation,

$\Rightarrow 2(2.5x^2 - 7x - 12) = 0 \\[1em] \Rightarrow 5x^2 - 14x - 24 = 0 \\[1em] \Rightarrow 5x^2 - 20x + 6x - 24 = 0 \\[1em] \Rightarrow 5x(x - 4) + 6(x - 4) = 0 \\[1em] \Rightarrow (5x + 6)(x - 4) = 0 \\[1em] \Rightarrow (5x + 6) = 0 \text{ or } (x - 4) = 0 \text{ ….[Using zero-product rule]}\\[1em] \Rightarrow x = \dfrac{-6}{5} \text{ or } x = 4 .$

(Since speed can’t be negative).

x = 4km/hr

Hence, x = 4km/hr.