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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.

\[ \Rightarrow \]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.

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