Question

# A train covered a certain distance at a uniform speed. If the train would have been 6 km/h faster, it would have taken 4 hours less than the scheduled time. And, if the train were slower by 6 km/h, it would have taken 6 hours more than the scheduled time. Find the length of the journey.

Hint: Distance in both cases will be equal. So, find equations of distance and both the cases and after that we can get the length of journey.
$\Rightarrow$Let the actual speed of the train be $x{\text{ km/hr}}$
$\Rightarrow$And the actual time taken be ${\text{y}}$ hours.
$\Rightarrow$As distance covered $=$ speed$*$time.
$\Rightarrow$So distance covered $= (xy){\text{km}}$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦...(1)
When the speed is increased by 6 km/hr,
Then time of journey is reduced by 4 hours.
So, speed will become,
Speed is $(x + 6){\text{km/hr}}$ and time of journey is $(y - 4)$hours.
So, distance covered will be,
$\Rightarrow$Distance covered $= {\text{ }}(x + 6)(y - 4).$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦...(2)
As we know that the distance covered will always be the same.
So, comparing equation 1 and 2. We get,
$\Rightarrow - 2x + 3y - 12 = 0$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦...(3)
When the speed is reduced by 6 km/hr,
Then time of journey is increased by 6 hours.
So, speed will become,
Speed is $(x - 6){\text{km/hr}}$ and time of journey is $(y + 6)$ hours.
So, distance covered will be,
$\Rightarrow$Distance covered $= {\text{ }}(x - 6)(y + 6).$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..(4)
As we know that distance covered will always be the same.
So, comparing equation 1 and 4. We get,
$\Rightarrow xy = (x - 6)(y + 6) = xy - 6y + 6x - 36$
From the above equation. We will get,
$\Rightarrow x - y - 6 = 0$ â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.(5)
Thus, we obtain following system of equations:
$\Rightarrow - 2x + 3y - 12 = 0 \\ \Rightarrow x - y - 6 = 0 \\$
By using cross-multiplication, we have,
$\Rightarrow \dfrac{x}{{(3)*( - 6) - ( - 1)*( - 12)}} = \dfrac{{ - y}}{{( - 2)*( - 6) - (1)*( - 12)}} = \dfrac{1}{{( - 2)*( - 1) - (1)*(3)}}$
On solving the above equation. It becomes,
$\Rightarrow \dfrac{x}{{ - 30}} = \dfrac{{ - y}}{{24}} = \dfrac{1}{{ - 1}}$
$\Rightarrow$So, $x = 30$ and $y = 24$
Putting the value of x and y in equation 1. We get,
Distance $= (30*24)km{\text{ }} = {\text{ }}720{\text{km}}$
$\Rightarrow$Hence, the length of the journey is 720km

Note: Whenever we came up with this type of problem then easiest and efficient way to find the length of journey is first, assume speed as a variable x and time as variable y and then make equations using formula, distance $=$speed$*$time. And then solve the equations by using a cross multiplication method to get the required length of journey.