Question

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?

Answer 1

`40 (km)/h`

Explanation:

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`

Answer 2

`40` km/h

Explanation:

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.