A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. What is the speed of the train?

Total Distance #S=480 km#
Formula For Speed
#"speed"="distance"/"time"#
Hence #"distance"="speed"xx"time"#

Case-I
let the uniform speed of Train #=v (km)/h#
and the time taken to complete distance #=(t ) "hour"#
Distance #S="speed"xx"time"=vt# . . . . . (equation 1)

Case-II
If speed has been 8km/h less then train would have taken 3 hours more to cover the same distance.
Now in this situation
speed of train #=(v-8) (km)/h#
and Time taken to complete distance #=(t+3) hour#
Distance #S="speed"xx"time"=(v-8)(t+3)# . . . . . (equation 2)

Since Distance for both cases are same .
Comparing equation 1 and equation 2, we get
#=>vt=(v-8)(t+3)#
#=>vt=(v)(t+3)-8(t+3)=vt+3v-8t-24#
cancel vt from both side
#=>0=3v-8t-24#
#=>3v-8t=24#
#=>3v=24+8t#
#=>v=(24+8t)/3#

but from equation 1 the value of #t=S/v=480/v (hour)#

#=>3v=24+8(480/v)#
#=>3v^2=24v+3840#
#=>3v^2-24v-3840=0#
#=>v^2-8v-1280=0#
factorize the quadratic equation
#=>(v-40)(v+32)=0#

#v# is either #40 (km)/h# or #-32 (km)/h#

Speed is Positive hence
#"Speed"=40 (km)/h#

Suppose the speed of the train is #x# km/h.
It takes #480/x# hours to travel, but if the speed were
8 km/h slower, it would take #480/(x-8)# hours.

Now we've got the equation:
#480/(x-8)# = #480/x# +#3#
This can be solved as follows.

1. Multiple both sides of the equation by #x(x-8)#.
   #480x = 480(x-8) +3x(x-8)#
2. Deform and factorize the equation to solve it.
   #x^2-8x-1280=0#
   #(x-40)(x+32)=0#
   #x=40, -32#

Don't forget that the answer must be positive, so the speed
is #x=40# km/h.

Let's check.
If we travel at 40 km/h, it takes 12 hours.
If we traveled at 32 km/h, it would take 15 hours, so the difference
is three hours and we've got the correct answer.