#### Question

A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/h more, the time taken for the journey would have been 1 hour 40 minutes less. Find the original speed of the car.

#### Solution

Let x km/h be the original speed of the car.

We know that,

Time taken = `"Distance"/"Speed"`

It is given that the car covers a distance of 400 km with the speed of x km/h.

Thus, the time taken by the car to complete 400 km is

t = `400/"x"`

Now, the speed is increased by 12 km.

Increased speed = (x + 12) km/h

Also given that, increasing the speed of the car will decrease the time taken by 1 hour 40 minutes.

Hence,

`400/"x" - 400/("x" + 12)` = 1 hour 40 minutes

⇒ `400/"x" - 400/("x" + 12)` = `1 40/60`

⇒ `(400 ("x" + 12) - 400"x")/("x"("x" + 12)) = 1 2/3`

⇒ `(400"x" + 4800 - 400"x")/("x"("x" + 12)) = 5/3`

⇒ `4800/("x"("x" + 12)) = 5/3`

⇒ 3 x 4800 = 5 × x × (x + 12)

⇒ 14400 = 5x^{2} + 60x

⇒ 5x^{2} + 60x - 14400 = 0

⇒ x^{2} + 12x - 2880 = 0

⇒ x^{2} + 60x - 48x - 2880 = 0

⇒ x (x + 60) - 48 (x + 60) = 0

⇒ ( x + 60) ( x - 48) = 0

⇒ x + 60 = 0 or x - 48 = 0

⇒ x = -60 or x = 48

Since, speed cannot be negative, we reject -60.

Hence, the original speed of the car is 48 km/h.