Train X completed the 100-mile trip in 5 hours
Speed = distance/time = 100/5 = 20 mph

Train Y completed the 100-mile trip in 3 hours
Speed = distance/time = 100/3 ≈ 33 mph (This approximation is close enough. You'll see why shortly)

How many miles had Train X traveled when it met Train Y?
Let's start with a word equation.
When the two trains meet, each train will have been traveling for the same amount of time
So, we can write: Train X's travel time = Train Y's travel time

time = distance/speed
We know each train's speed, but not the distance traveled (when they meet). So, let's assign some variables.

Let d = the distance train X travels
So, 100-d = the distance train Y travels (since their COMBINED travel distance must add to 100 miles)

We can now turn our word equation into an algebraic equation.
We get: d/20 = (100 - d)/33
Cross multiply to get: (33)(d) = (20)(100 - d)
Expand: 33d = 2000 - 20d
Add 20d to both sides: 53d = 2000
So, d = 2000/53

IMPORTANT: Before you start performing any long division, first notice that 2000/50 = 40
Since the denominator is greater than 50, we can conclude that 2000/53 is LESS THAN 40
Since only one answer choice is less than 40, the correct answer must be A

Here's another approach....

speed = distance/time
So, Train X's speed = 100/5 = 20 miles per hour
And Train Y's speed = 100/3 ≈ 33 miles per hour

So the shrink rate = 20 + 33 = 53 miles per hour.

At the beginning, the two trains are 100 miles apart.
time = distance/speed
So the time it takes for the trains to meet (i.e., the time it takes for the distance between the two trains to shrink from 100 miles to 0 miles) = 100/53 = a little bit less than 2 hours

How many miles had Train X traveled when it met Train Y?
Distance = (speed)(time)
So Train X's distance traveled = (20)(a little bit less than 2 hours) = a little bit less than 40 miles

Answer: A

Cheers,
Brent

I did something a bit different, here. I calculated their gap-closing average (20mph + 100/3mph) to get 160/3mph combined, then simply set the distance to 100 = rate of 160/3 times time (T). Once that was completed, I got Total Time as 1.875.

Since I'm not great with algebra, this was helpful because I could simply plug in answer choices until I found 1.875 hours!

We can solve it through relative speed concept,

X : 100-mile trip in 5 hours thus , speed of X in 1hr= 100/5

Y : 100-mile trip in 3 hours thus , speed of Y in 1hr= 100/3

Distance between them is 100

Total Meeting Time = 100 / 100/5 + 100/3 = 15/8 hr

X Distance travel = speed * time = 100/5 * 15/8 = 300/8 = 37.5 thus A