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that wasn't funny. is this representative of spoiler alert as a whole?


Two trains leave different cities heading toward each other at different speeds. When and where do they meet?

Train A, traveling 70 miles per hour (mph), leaves Westford heading toward Eastford, 260 miles away. At the same time Train B, traveling 60 mph, leaves Eastford heading toward Westford. When do the two trains meet? How far from each city do they meet?

To solve this problem, we'll use the distance formula:



Distance = Rate x Time


Since an equation remains true as long as we perform the same operation on both sides, we can divide both sides by rate or by time.


So rate is defined as distance divided by time, which is a ratio.

Speed is another word that is used for rate. When a problem says that a train is moving at a speed of 40 mph, you can understand this to mean that the train's rate is 40 mph, which means it will travel 40 miles in one hour.


Here are two different ways to approach this problem. Let's start by listing the information given:


Speed of Train A: 70 mph
Speed of Train B: 60 mph
Distance between Westford and Eastford: 260 miles


Method I: We'll use the notion of relative speed 1 (or relative rate) in order to express the rates of the two trains in one number that can then be used in the distance formula.

Imagine you're on Train A. You're going 70 mph, so your speed relative to the trees, houses, and other non-moving things outside the train is 70 mph. (All of those objects look as if they're going by at 70 mph.) Now imagine you're the engineer and you can see Train B coming toward you - not on the same track, of course! Since Train B is moving 60 mph, it will look as if it's approaching faster than if it were sitting still in the station - a lot faster than the trees and houses appear to be moving.

The relative speed of the two trains is the sum of the speeds they are traveling. (If you're on either of the trains, this is the speed you appear to be moving when you see the other train.) In our problem, the relative speed of the two trains is 70 mph + 60 mph = 130 mph. What if the trains were traveling in the same direction? Then we'd need to subtract the speed of the slower train from the speed of the faster train, and their relative speed would be 10 mph.

At this point we know two of the three unknowns: rate and distance, so we can solve the problem for time. Remember that time = distance/rate, the distance traveled is 260 miles, and the relative speed is 130 mph:


t = 260 miles/130 mph
t = 2 hrs.


We find that the trains meet two hours after leaving their respective cities.