It takes the high-speed train x hours to travel the z miles

We have a converging rate problem in which:

Distance(1) + Distance(2) = Total Distance

We are given that 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. Thus, we know the following:

rate of the high-speed train = z/x

rate of the regular speed = z/y

We are also given that they leave at the same time, so we can let the time when both trains pass each other be t. We can substitute our values into the total distance formula.

Distance(1) + Distance(2) = Total Distance

(z/x)t + (z/y)t = z

zt/x + zt/y = z

We can divide the entire equation by z and we have:

t/x + t/y = 1

Multiplying the entire equation by xy gives us:

ty + tx = xy

t(y + x) = xy

t = xy/(y + x)

Now we can calculate the distance traveled by both trains for time t, using the formula

distance = rate x time

distance of high speed train = (z/x)[xy/(y + x)]

distance of regular speed train = (z/y)[xy/(y + x)]

Now we need to calculate the difference between the distance of the high speed train and the distance of the regular speed train:

difference of distance traveled = distance of high speed train - distance of regular speed train

difference of distance traveled = (z/x)[xy/(y + x)] - (z/y)[xy/(y + x)]

We can factor out [xy/(y + x)]:

difference of distance traveled = [xy/(y + x)](z/x - z/y)

difference of distance traveled = [xy/(y + x)][(yz - xz)/(xy)]

The xy terms cancel and we are left with:

difference of distance traveled = (yz - xz)/(y + x) = z(y-x)/(x + y)

Answer: A