Atrain travelled a certain distance at a uniform speed. Had the speed been 6 km per hour more, it would have needed 4 hours less. Had the speed been 6 km per hour less, it would have needed 6 hours more. The distance, in km, travelled by the train is:

The correct option is 720.

Let the uniform speed of the train be u km/h and the scheduled time to cover the distance be t hours.
The distance travelled by the train is given by the formula:
$$D=u\times t$$

Case 1: If the speed had been 6 km/h more, the time taken would be 4 hours less.
New speed = $(u+6)$ km/h
New time = $(t-4)$ hours
Since the distance remains the same, we can write:
$$D=(u+6)(t-4)$$ Substituting D = ut into the equation:
$$ut=ut-4u+6t-24$$ Simplifying this, we get:
$$-4u+6t=24$$ Dividing by 2:
$$-2u+3t=12\text{(Equation 1)}$$

Case 2: If the speed had been 6 km/h less, the time taken would be 6 hours more.
New speed = $(u-6)$ km/h
New time = $(t+6)$ hours
Again, since the distance is constant:
$$D=(u-6)(t+6)$$ Substituting D = ut into the equation:
$$ut=ut+6u-6t-36$$ Simplifying this, we get:
$$6u-6t=36$$ Dividing by 6:
$$u-t=6\text{(Equation 2)}$$

Solving the system of linear equations:
From Equation 2, we can express u in terms of t:
$$u=t+6$$ Substitute this expression for u into Equation 1:
$$-2(t+6)+3t=12$$ $$-2t-12+3t=12$$ $$t=24$$ So, the scheduled time taken is 24 hours.
Now, find the uniform speed u:
$$u=24+6=30$$ The speed of the train is 30 km/h.

Calculating the distance travelled:
$$D=u\times t=30\times 24=720\text{km}$$ Thus, the distance travelled by the train is 720 km.