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" ?

Answer 1

`4`.

Explanation:

This is simple equation problem.

The velocity for half the distance = `x` km/h.

So, Time taken to cross the first half = `(12/2)/x` hours.

The velocity for the other half = `x + 2` km/h.

So, Time taken to cross the rest = `(12/2)/(x + 2)` hours.

And, `2` hours and `30` minutes = `2 1/2` hours.

So, On with the equation.

`color(white)(xxx)(12/2)/x + (12/2)/(x+2) = 2 1/2`

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

`rArr 6(1/x + 1/(x + 2)) = 5/2` [Taking the 6 as common part]

`rArr ((x + 2) + x)/(x(x +2)) = 5/2 * 1/6` [Transposed the 6]

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

`rArr (2x + 2)/(x^2 + 2x) = 5/12` [Simplified using L.C.M]

`rArr 24x + 24 = 5x^2 + 10x` [Cross multiplication]

`rArr 5x^2 + 10x - 24x - 24 = 0` [Transposing everything to a side of the equation]

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

Now, Use The Sridhar Acharya's Rule or Quadratic Formula to solve this.

And, Don't forget to check the Discriminant first.

`D = b^2 - 4ac = (-14)^2 - 4 * 5 * (-24) = 196 + 480 = 676 gt 0`

So, Definitely There are two real solutions for `x` and both are distinct.

Now first solution for `x` = `(- b + sqrt(D))/(2a) =( -(-14) + sqrt(676))/(2 * 5)`

`color(white)(xxxxxxxxxxxxxx)= (14 + 26)/10 = 40/10 = 4`

Now, The second solution for `x`

= `(- b - sqrt(D))/(2a) =( -(-14) - sqrt(676))/(2 * 5)`

`color(white)(xxxxxxxxxxxxxx)= (14 - 26)/10 = -12/10 = -1.2`

So, `x = 4, -1.2`.

But speed can't be negative in any circumstances as distance travelled can't be negative.

So, `x = 4` is the legitimate value.