Two trains, X and Y, started simultaneously from opposite ends of a 100-mile route and traveled toward each other on parallel tracks. Train X, traveling at a constant rate, completed the 100-mile trip in 5 hours; Train Y, traveling at a constant rate, completed the 100-mile trip in 3 hours. How many miles had train X traveled when it met train Y?

(A) 37.5
(B) 40.0
(C) 60.0
(D) 62.5
(E) 77.5
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Ajit

Train A's Speed 100/5 == 20 miles/hr
B's Speed 100/3 = 33.3 miles/hr

Relative speed 53.3 miles/hr

To cover 100 miles with relative speed time taken 100/53.3 almost 135 % of an hour

Train A will cover 135% of 20 approx 37.6

HTH

The concept used in these questions is Relative Speed.

If two people walk in opposite directions (either towards each other or away from each other), their speed relative to each other is the sum of their speeds. e.g. If you are walking away from me at a speed of 2 miles/hr and I am walking away from you at a speed of 1 mile/hr, together we are creating a distance of 3 miles in 1 hr between us so our relative speed is 2 + 1 = 3 miles/hr
On the other hand, when two people walk in the same direction, their relative speed is the difference between their speeds.
e.g. if you are walking away from me at 1 mile/hr and I am walking towards you at 2 miles/hr, my speed relative to you is 2-1 = 1 mile/hr.

Time taken to meet = Total distance traveled/Relative speed

Speed of train X = 100/5 = 20 miles/hr
Speed of train Y = 100/3 miles/hr
Relative Speed = 20 + 100/3 = 160/3 miles/hr
Distance between them = 100 miles
Time taken to meet = 100/(160/3) hr = 15/8 hrs

In this time, train X would have traveled 20 * (15/8) = 37.5 miles

Faster Alternate Approach using Ratios :

Time taken by train X : Time taken by train Y = 5:3
Then, Speed of train X:Speed of train Y = 3:5
Since they start simultaneously, they travel for same time. So the ratio of their distance covered should be same as ratio of their speeds.
Distance covered by train X : Distance covered by train Y = 3:5
3/8 *100 = 37.5 miles (Distance covered by train X)
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As I can see, I notice different approachs that are good but not so efficient

relative rates : 5 + 3 = 8

RT= D----> \frac{100}{8} = 12.5

Now this is a problem where the 2 times are dissimilar, is like a weigthed average in some how: what is the train that have more weigth in this scenario: the train with the 3 hours of trip.

So : 12.5 * 3 = 37.5

A is the answer. This is the fastest approach you can do with these tricky problems. that's it
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KUDOS is the good manner to help the entire community.