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" ?

1 Answer

4.

Explanation:

Soumalya PramanikSoumalya Pramanik

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.