Question

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Answer 1

Hint:In this question, we use the formula of speed-distance relation. First we have to make two equations according to the given question in which two variables speed and time then solve both the equations. So, we will get the required answer.

Complete step-by-step answer:

Let the original speed of the train be v km/h.
A train travels 360 km at a speed of v km/h then we have to find time taken by train to cover 360km.
Time taken to cover a distance of 360 km \[ = \dfrac{{{\text{distance}}}}{{{\text{speed}}}}\]
$t = \dfrac{{360}}{v}............\left( 1 \right)$
Now, the speed of the train increases by 5 km/h so it would have taken 1 hour less to cover 360km.
$t - 1 = \dfrac{{360}}{{v + 5}}................\left( 2 \right)$
Now, subtract (2) equation from (1) equation.
$
   \Rightarrow \left( t \right) - \left( {t - 1} \right) = \dfrac{{360}}{v} - \dfrac{{360}}{{v + 5}} \\
   \Rightarrow t - t + 1 = \dfrac{{360}}{v} - \dfrac{{360}}{{v + 5}} \\
   \Rightarrow 1 = \dfrac{{360}}{v} - \dfrac{{360}}{{v + 5}} \\
 $
Now, we have only one variable so we can easily solve it.
$
   \Rightarrow 1 = \dfrac{{360\left( {v + 5 - v} \right)}}{{v\left( {v + 5} \right)}} \\
   \Rightarrow 1 = \dfrac{{360 \times 5}}{{v\left( {v + 5} \right)}} \\
 $
Cross multiplication,
$
   \Rightarrow v(v + 5) = 360 \times 5 \\
   \Rightarrow {v^2} + 5v - 1800 = 0 \\
 $
We can see the quadratic equation in v. So, we can solve the quadratic equation by splitting the middle term.
\[
   \Rightarrow {v^2} + 5v - 1800 = 0 \\
   \Rightarrow {v^2} + 45v - 40v - 1800 = 0 \\
   \Rightarrow v\left( {v + 45} \right) - 40(v + 45) = 0 \\
   \Rightarrow \left( {v + 45} \right)\left( {v - 40} \right) = 0 \\
   \Rightarrow v = - 45,40 \\
\]
We know speed cannot be negative. So, we eliminate v=-45 km/h.
So, the speed of the train is 40km/h.

Note: Whenever we face such types of problems we use some important points. Like first we make a quadratic equation by eliminating one variable from two equations. So, after solving the quadratic equation we will get the required answer.