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

Given: Distance travelled by the train, d=180km, increased speed of the train=9km/h and less time taken by the train=1 hour.

To do: To find the speed of the train.

Solution: Let us say that the train travels from A to B with a uniform speed s km/h and time taken by the train for the journey is t hours.

As known that $distance=speed\times time$ or $time=\frac{distance}{speed} $ or $t=\frac{d}{s}$

The train travelled $d=180\ km$

$\therefore \ t=\frac{180}{s}$ $\dotsc \dotsc \dotsc \dotsc \dotsc ( 1)$

On increasing its speed 9 km/h, it takes 1 hour less time to reach the same destination, as given in the question

$t-1=\frac{180}{s+9}$

or

$\ t=\frac{180}{s+9} +1$ $\dotsc \dotsc \dotsc \dotsc \dotsc \dotsc ( 2)$

on comparing $( 1)$ and $( 2)$

$t=\frac{180}{s} =\frac{180}{s+9} +1$

Or

$s( 189+s) =180( s+9)$

$\Rightarrow s^{2} +189s=180s+1620$

$\Rightarrow s^{2} +9s-1620=0$

$\Rightarrow ( s-36)( s+45) =0$

If $s+45=0$

$\Rightarrow s=-45$

$\because$ speed can't be negative,we reject this value.

If $s-36=0$

$\Rightarrow s=36$ km/h

$\therefore$ The speed of the train is 36 km/h.

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