Manhattan Test Problem - Why time is taken as constant?

So, they take same time by the time they meet.

bubbliiiiiiii wrote:It takes the high-speed train x hours to travel the z miles from Town A to Town B at a constant rate, while it takes the regular train y hours to travel the same distance at a constant rate. If the high-speed train leaves Town A for Town B at the same time that the regular train leaves Town B for Town A, how many more miles will the high-speed train have traveled than the regular train when the two trains pass each other?


z(y – x)
---------
x + y



z(x – y)
---------
x + y



z(x + y)
---------
y – x



xy(x – y)
---------
x + y



xy(y – x)
---------
x + y

Solution:

Since the trains traveled the z miles in x and y hours, their speeds can be represented as z/x and z/y respectively.

We can again use an RTD chart to evaluate how far each train travels when they move toward each other starting at opposite ends. Instead of using another variable d here, let's express the two distances in terms of their respective rates and times.



High-speed Regular Total

R z/x z/y

T t t

D zt/x zt/y z


Since the two distances sum to the total when the two trains meet, we can set up the following equation:

zt/x + zt/y = z divide both sides of the equation by z
t/x + t/y = 1 multiply both sides of the equation by xy
ty + tx = xy factor out a t on the left side
t(x + y) = xy divide both sides by x + y
t =
xy

x + y


To find how much further the high-speed train went in this time:
(ratehigh × time) – (ratereg × time)
(ratehigh – ratereg) × time

zy – zx xy
------- x ------
xy x+y


z(y – x)
-------
x + y

The correct answer is A.

bubbliiiiiiii wrote:Since the two trains are travelling at different speeds what I understand is that the slow moving train would need more time to cover the same distance that is travelled by fast moving train.

However, we can simply answer the question saying that when two elements among three in formula speed = distance/time, are variables the third has to be treated as constant. But, I want to understand, why is it constant?

Hope this makes the question more clear.