Question

A train travels 180 km at a uniform speed. If the speed had been 9 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train. [CBSE 2013]

Answer 1

Distance covered by the train = 180 km
We know that distance covered$\left(d\right)=\mathrm{speed}\left(s\right)\times \mathrm{time}\left(t\right)$
$$\begin{gathered} \Rightarrow s\times t=180 \\ \Rightarrow s=\frac{180}{t}...\left(\mathrm{i}\right) \end{gathered}$$
Also, given that if the speed is increased by 9km/h, time of travel gets reduced by 1 hour.
$\Rightarrow (s+9)(t-1)=180...\left(\mathrm{ii}\right)$
Put the value of (i) in (ii), we get
$$\begin{gathered} \left(\frac{180}{t}+9\right)(t-1)=180 \\ (180+9t)(t-1)=180t \\ 180t-180+9t^2-9t=180t \\ 9t^2-9t-180=0 \\ {t}^2-t-20=0 \\ {t}^2-5t+4t-20=0 \\ t(t-5)+4(t-5)=0 \\ (t+4)(t-5)=0 \\ (t+4)=0\mathrm{or}(t-5)=0 \\ t=-4\mathrm{or}t=5 \end{gathered}$$
Ignore the negative value
So, time taken = 5 hours
From (i)
$s=\frac{180}{t}=\frac{180}{5}=36$
Hence, the speed is 36 km/h.