Question 737280
A van and a truck left city a and b respectively at the same time traveling toward each other.
 When they met, the truck had traveled 120 more miles than the van.
 Afterwards, the van traveled 9 more hours to reach city B while the truck traveled 4 more hours to reach city A. 
 What is the distance between the 2 cities?
:
let x = distance from a to meeting point (initially traveled by the van)
then
(x+120) = distance from b to meeting point (initially traveled by the truck)
therefore
(2x+120) = distance from a to b
:
Speed = dist/time
{{{(x+120)/9}}} = speed of the van traveling to b
{{{x/4}}} = speed of the truck traveling to a
:
Let t = travel time for both to the meeting point (time = dist/speed)
Truck time
{{{(x+120)/(x/4)}}} = (x+120)*{{{4/x}}} = {{{(4x+480)/x}}}
Van time
{{{x/((x+120)/9)}}} = x * {{{9/(x+120)}}} = {{{(9x)/(x+120)}}}
Times are equal, therefore
{{{(4x+480)/x}}} = {{{(9x)/(x+120)}}}
cross multiply
9x^2 = (4x+480)(x+120)
FOIL
9x^2 = 4x^2 + 480x + 480x + 57600
0 = 4x^2 - 9x^2 + 960x + 57600
-5x^2 + 960x + 57600 = 0
simplify, divide by-5
x^2 - 192x - 11520 = 0
Factors to
(x-240)(x+48) = 0
The positive solution
x = 240
a to b distance; 2(240) + 120 = 600 miles
:
:
We can confirm this, find the speed of each
240/4 = 60 km/hr the truck
360/9 = 40 km/hr the van
:
Time to meet point should be the same
truck: 360/60 = 6 hrs
van: 240/40 = 6 hrs