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

Let the usual speed of the train be x \mathrm{~km} / \mathrm{h}.

\therefore Reduced speed of the train =(x-8) \mathrm{km} / \mathrm{h}

Total distance to be covered =480 \mathrm{~km}

Time taken by the train to cover the distance at usual speed
=\frac{480}{x} h \quad\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} 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 \mathrm{~h}

\begin{array}{l} \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(x-40)+32(x-40)=0 \\ \Rightarrow(x-40)(x+32)=0 \\ \Rightarrow x-40=0 \text { or } x+32=0 \\ \Rightarrow x=40 \text { or } x=-32 \end{array}

\therefore x=40

(Speed cannot be negative)

Hence, the usual speed of the train is 40 \mathrm{~km} / \mathrm{h}.