Last week, in Number Sense 024, we took a look at modeling a simple subtraction, to find out how to produce an equation from a word problem. We ended that essay with a warning that this week we would take up that classic word problem of two trains passing in the night.
A fast train leaves New York at 6 pm and arrives in Boston at 9 pm. A slower train leaves Boston an hour after the first train leaves, and arrives in New York at midnight. If both trains travel non-stop at constant speed, what time do the trains pass each other?
One thing the problem doesn't tell us seems like it would be important information: how far is it from Boston to New York? We can assume that both trains are traveling similar distances (the Boston bound train, for example, is not going to Boston via Pittsburgh.) But we don't know what that distance is.
Here is a simple model of one train's trip from New York to Boston
The red arrow represents the train's trip. At 6pm it is in New York, at 7pm it is 1/3 of the way to Boston, at 8pm it is 2/3s of the way, and it arrives in Boston at 9pm. I don't know how many miles the train traveled, so I just labeled the beginning (at the bottom of the rectangle) “New York” and the end (at the top) “Boston”
Likewise, we can make another model for the train traveling from Boston to New York.
The Boston/New York train leaves Boston an hour later, and doesn't get to New York until midnight.
If we superimpose those two pictures, we get this
Which show us that the trains pass each other sometime between 7pm (when the Boston train leaves Boston) and 9pm (when the New York train arrives at Boston.)
Now, the height of those rectangles represent the distance the train travels. We have made an assumption about how far that is by making the rectangles a certain height. What happens if our assumption is wrong? What if the distance is greater or less than we have shown it?
Well, here is a picture showing three rectangles of different heights, that is to say, different distances between New York and Boston. The times, or widths, of the rectangles are the same, because we know what time the trains left and arrived. If we take a look at what time those two trains passed each other,
we can see that it doesn't really matter how far apart the two cities are. If they are further apart, both trains run faster to leave and arrive on time, if they are closer together, the trains run slower. In any case, they pass each other at the same time.
So it turns out it isn't important (for solving this problem) to know the distance from Boston to New York. The only thing we need to know is that it is the same distance as New York to Boston.
A fast train leaves New York at 6 pm and arrives in Boston at 9 pm. A slower train leaves Boston an hour after the first train leaves, and arrives in New York at midnight. If both trains travel non-stop at constant speed, what time do the trains pass each other?
We know the fast train travels the distance in 3 hours, and the slow train travels the distance in 5 hours.
The speed of a train is the distance it travels divided by the time it takes to travel that distance. If a train travels 50 mph for one hour, it covers 50 miles. If it goes for three hours, it covers 150 miles. If we divide 150 miles by 3 hours, we get 50 mph. Speed is distance per unit of time. If we travel for more than one unit of time (hours, minutes, days, whatever) then we must divide by the number of time units.
Speed = distance / time. (100 miles in two hours = 50 miles per hour)
That is part of a multiplication fact family. There are two multiplications and one other division in the family. They are:
Time = distance / speed (100 miles at 50 miles per hour takes two hours)
Distance = speed x time (two hours at 50 miles per hour is 100 miles)
and Distance = time x speed (since multiplication is commutative)
The first equation, Speed = distance / time, is the definition of speed. The remaining members of this fact family allow us to calculate any of the three amounts, if we know the other two.
So, we know the time the trains took. We don't know the distance, but we do know the distance doesn't affect the answer to this problem, so we can replace it with some arbitrary number. Let's call it “d”.
Since the slow train started later, we will say it was traveling for t hours when the fast train passed it.
At that time, the fast train had been traveling for t+1 hour, since it started an hour earlier.
When the trains pass each other, the distance the fast train traveled plus the distance the slow train traveled adds up to the total distance between New York and Boston.
Let's use the distributive property first
Then multiply both sides by 15 to get rid of the denominators on the right side of the equation
15d = 5td + 5d + 3td
Divide both sides by d
15 = 5t + 5 + 3t
Subtract 5 from both sides
10 = 5t + 3t
Add the right side
10 = 8t
Divide by 8
so, the trains passed 10/8ths of an hour past 7pm. That's the same as 1 hour and 2/8ths, or 1 hour and a quarter, or 1 hour 15 minutes past 7pm, or 8:15pm.
I seem to have bit off a bit more than I should have by promising to visit the notorious trains problem so soon. It seems to me I'm making a lot of unexplained leaps in this exposition, big lumps in the porridge, so to speak, instead of the bite sized chucks I usually publish.
What do you think? Have fun in the comments.