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. [CBSE 2013C]

Answer 1

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 = $\frac{480}{x}$ h $\left(\mathrm{Time}=\frac{\mathrm{Distance}}{\mathrm{Speed}}\right)$

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 h

$$\begin{gathered} \therefore \frac{480}{x-8}=\frac{480}{x}+3 \\ \Rightarrow \frac{480}{x-8}-\frac{480}{x}=3 \\ \Rightarrow \frac{480x-480x+3840}{x\left(x-8\right)}=3 \\ \Rightarrow \frac{3840}{x^2-8x}=3 \end{gathered}$$
$$\begin{gathered} \Rightarrow x^2-8x=1280 \\ \Rightarrow x^2-8x-1280=0 \\ \Rightarrow x^2-40x+32x-1280=0 \\ \Rightarrow x\left(x-40\right)+32\left(x-40\right)=0 \end{gathered}$$
$$\begin{gathered} \Rightarrow \left(x-40\right)\left(x+32\right)=0 \\ \Rightarrow x-40=0\mathrm{or}x+32=0 \\ \Rightarrow x=40\mathrm{or}x=-32 \end{gathered}$$
∴ x = 40 (Speed cannot be negative)

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