#### Question

A girl goes to her friend’s house, which is at a distance of 12 km. She covers half of the distance at a speed of x km/hr and the remaining distance at a speed of (x + 2) km/hr. If she takes 2 hrs 30 minutes to cover the whole distance, find ‘x’.

#### Solution

We know

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

Given the girl covers a distance of 6 km at a speed x km/hr

Time taken to cover first 6 km = `6/"x"`

Also the girl covers the remaining 6 km distance at a speed (x + 2) km/hr.

Time taken to cover next 6 km = `6/("x" + 2)`

Time taken to cover the whole distance = 2 hrs 30 mins - 2 `30/60 = 2 1/2 = 5/2` hrs

`∴ 6/x + 6/(x + 2) = 5/2`

`(6x + 12 + 6x)/(x(x + 2)) = 5/2`

`(12 + 12x)/(x^2 + 2x) = 5/2`

`24 + 24x = 5x^2 + 10x`

`5x^2 - 14x - 24 = 0`

`5x^2 - 20x + 6x - 24 = 0`

5x(x - 4) + 6(x - 4) = 0

(5x + 6)(x - 4) = 0

`x = (-6)/5, 4`

Since speed cannot be negative. Therefore x = 4