Question

A car X left town A for town B at the same time that another car Y left town B for town A, each travelling at constant speed. They met 80 minutes later and car X arrived at town B 36 minutes after car Y reached town A. How long did it take car X to reach town B?

Answer 1

Let x = speed of car X
y = speed of car Y
L = distance of AB

"meet 80minutes later"
L = 80x + 80y (distance = velocity * time)

The time for car X to reach town B is:
tx = ty + 36

But, tx = L/x and ty = L/y

Substitute,
tx = ty + 36
[80(x+y)]/x = [80(x+y)]/y + 36

Multiplying both sides by xy
80(x+y)y = 80(x+y)x + 36xy ~ 20(x+y)y - 20(x+y)x = 9xy

Expand:
20y^2 - 20x^2 - 9xy = 0
Divide both sides by x^2, and let k=y/x
20k^2 - 9k - 20 = 0
k = 5/4, so x = 4 and y= 5

Substitute values to the equation tx
tx = 180mins.

Is this correct? Thank you.

Answer 2

So, did you actually check your answer?
tx=180 means ty=144
So, total time for X is 180+80=260
total time for Y is 144+80 = 224
distance from A of X to meeting place is L/260*80
distance from B of Y to meeting place is L/224*80
Add those up and you do not get L.


Your mistake is here:
But, tx = L/x and ty = L/y
tx is not L/x. tx is the time X took after the initial 80 minutes. Not sure how you can fit that into your equations, but I did it like this:

If y = kx, then in the first 80 minutes, L=80x+80y = 80x(1+k)

(80x(1+k)-80x)/x = (80x(1+k)-80kx)/kx + 36
80(1+k)-80 = (80(1+k)-80k)/k + 36
80k^2 = 80+36k
20k^2-9k-20 = 0
k = 5/4

So, Y travels 5/4 as fast as X.

So, in the 1st 80 minutes, X went 4/9 of the way to B and Y went 5/9 of the way to A.

So, X took 5/4 as long to make the rest of the trip, or another 100 minutes.

Not sure whether the question is X's total time (180 min) or remaining time (100 min)

Check:
Y took 4/5 as long to get to A as it did to get to the meeting place. That's 64 minutes. Y's total time = 144 minutes.

180 = 144+36

Answer 3

From what I understood in the problem, the required is the total travel time of car X from town A to town B.

In the 180minutes, I think that the "80minutes later" is already included so no need to add it up.