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

Answer:

#4#.

Explanation:

Soumalya 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.