A train covers a distance of 480 km at a uniform speed.
Question:

A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/h less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.

Solution:

Let the usual speed of the train be x km/h.

∴ Reduced speed of the train = (x − 8) km/h

Total distance to be covered = 480 km

Time taken by the train to cover the distance at usual speed $=\frac{480}{x} \mathrm{~h}$        $\left(\right.$ Time $\left.=\frac{\text { Distance }}{\text { Speed }}\right)$

Time taken by the train to cover the distance at reduced speed $=\frac{480}{x-8} \mathrm{~h}$

According to the given condition,

Time taken by the train to cover the distance at reduced speed = Time taken by the train to cover the distance at usual speed + 3 h

$\therefore \frac{480}{x-8}=\frac{480}{x}+3$

$\Rightarrow \frac{480}{x-8}-\frac{480}{x}=3$

$\Rightarrow \frac{480 x-480 x+3840}{x(x-8)}=3$

$\Rightarrow \frac{3840}{x^{2}-8 x}=3$

$\Rightarrow x^{2}-8 x=1280$

$\Rightarrow x^{2}-8 x-1280=0$

$\Rightarrow x^{2}-40 x+32 x-1280=0$

$\Rightarrow x(x-40)+32(x-40)=0$

$\Rightarrow(x-40)(x+32)=0$

$\Rightarrow x-40=0$ or $x+32=0$

$\Rightarrow x=40$ or $x=-32$

∴ x = 40              (Speed cannot be negative)

Hence, the usual speed of the train is 40 km/h.