# 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?

Sep 27, 2017

$40 \frac{k m}{h}$

#### Explanation:

Total Distance $S = 480 k m$
Formula For Speed
$\text{speed"="distance"/"time}$
Hence $\text{distance"="speed"xx"time}$

Case-I
let the uniform speed of Train $= v \frac{k m}{h}$
and the time taken to complete distance $= \left(t\right) \text{hour}$
Distance $S = \text{speed"xx"time} = v t$ . . . . . (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 $= \left(v - 8\right) \frac{k m}{h}$
and Time taken to complete distance $= \left(t + 3\right) h o u r$
Distance $S = \text{speed"xx"time} = \left(v - 8\right) \left(t + 3\right)$ . . . . . (equation 2)

Since Distance for both cases are same .
Comparing equation 1 and equation 2, we get
$\implies v t = \left(v - 8\right) \left(t + 3\right)$
$\implies v t = \left(v\right) \left(t + 3\right) - 8 \left(t + 3\right) = v t + 3 v - 8 t - 24$
cancel vt from both side
$\implies 0 = 3 v - 8 t - 24$
$\implies 3 v - 8 t = 24$
$\implies 3 v = 24 + 8 t$
$\implies v = \frac{24 + 8 t}{3}$

but from equation 1 the value of $t = \frac{S}{v} = \frac{480}{v} \left(h o u r\right)$

$\implies 3 v = 24 + 8 \left(\frac{480}{v}\right)$
$\implies 3 {v}^{2} = 24 v + 3840$
$\implies 3 {v}^{2} - 24 v - 3840 = 0$
$\implies {v}^{2} - 8 v - 1280 = 0$
$\implies \left(v - 40\right) \left(v + 32\right) = 0$

$v$ is either $40 \frac{k m}{h}$ or $- 32 \frac{k m}{h}$

Speed is Positive hence
$\text{Speed} = 40 \frac{k m}{h}$

Sep 27, 2017

$40$ km/h

#### Explanation:

Suppose the speed of the train is $x$ km/h.
It takes $\frac{480}{x}$ hours to travel, but if the speed were
8 km/h slower, it would take $\frac{480}{x - 8}$ hours.

Now we've got the equation:
$\frac{480}{x - 8}$ = $\frac{480}{x}$ +$3$
This can be solved as follows.

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

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.