It’s a trope on TV and stand-up comedy by this time: a person is tasked with applying mathematics skills by considering when a train leaving Ypsilanti at 9:00 a.m., traveling at 50 miles per hour, will cross paths with a train leaving from other city further away at a different time and traveling at a different speed.
But inasmuch as Jerry Seinfeld never said “What is the deal with airline peanuts” and Elsa never said “Play it again, Sam,” was this ever actually a thing in any math textbook printed in English? Or is it a giant Mandela Effect?
Yes word problems involving moving vehicles are pretty common in math texts. Picking the first highschool text that came up on google books and searching for train produced this which involves a train and a fly. I would expect problems involving two trains are even more common.
It did not take me much googling to find a math textbook from 1920. Click here and then click on “Click here to download Everyday Arithmetic Book 2 (grades 5 & 6)”. You’ll find pages 170-171 interesting, particularly question 5 on page 170, and questions 8-9 on page 171. It’s not exactly what the OP specified, but it is close enough to convince me that those questions really did exist.
(Note: When I said “pages 170-171”, I was referring to the printed page number, which corresponds to pages 180-181 of the PDF.)
Yes, I’ve seen such math problems on K-12 textbooks. They were the worst; I still cannot solve them to this day.
One of my favorite “analogies made by high schoolers” is the one that reads: *“The two star-crossed lovers raced towards each other under the moonlit sky, one like a train leaving Denver at 7:30 PM at 45 miles per hour and the other like a train leaving San Francisco at 8:00 PM at 55 miles per hour.” *
We had a variety of problems like that when I took Algebra I, in 9th grade. They were called “Uniform Motion Problems”. I don’t recall ever seeing a problem like that before then, and I don’t know that I would have had any idea how to solve them before First Semester Algebra.
I recall this, though: In that Algebra I class, NOBODY could understand how to do those problems, including me. There was just something about them that we couldn’t get. I recall that we must have spent a whole week on them, that should have been just maybe two days, and the teacher finally said we had to move on.
Fast forward a semester: Algebra II, started with a review of a bunch of problems from Algebra I, including some of those. Suddenly, they seemed trivial, to me at least. I couldn’t understand why I ever had so much trouble with them before.
Ah, yes, it’s coming back to me, vague and shadowy through the mists of time.
These problems were supposed to give you practice working with the
Distance = Rate x Time
formula.
The catch was, you had two trains (or boats or airplanes or star-crossed lovers or whatever). Each of them had its own distance, its own rate, and its own time. So you have TWO separate D = R x T equations to work with. So there were a total of six numbers in the two equations. The problems had many variations, but you were always given some of those six numbers and you had to find the missing numbers in the two D = R x T equations.
But neither equation alone had enough information. You had to somehow put together the information given in the two equations to develop enough information to solve for all the missing number(s) in both equations.
The trick was, there was always something that was the same in both of the equations. Either the trains both traveled the same distance (both D’s the same), or they both traveled the same speed (both R’s the same) or both for the same length of time (both T’s the same), or there were further variations on that. When you figured that out, you have the information you needed to combine the two equations into one equation that you could solve.
NO algebra textbook or teacher that I know of EVER explained it quite the way I just did above. We all had to figure that out for ourselves. Those who did, could eventually learn to solve those problems. Those who didn’t, couldn’t.
In the long-lost days of my youth, from the earliest days I can remember, there was a old college algebra textbook laying around the house. It somehow gravitated into my bedroom and I still have it to this very day. (I’m 68 years old now.) I just dug it out of a dusty cardboard box at the bottom of my closet.
It’s called Algebra For College Students by Jack Britton and L. Clifton Snively, and printed in 1948. That makes it a few years older than me. It’s a much better algebra textbook than any other I’ve seen. (And, as a math major and occasional tutor, I’ve seen a few. Yes, IMHO, just as often alleged, they got more and more dumbed down every year.) It must have been a popular textbook in its day, because I’ve seen it in used book stores from time to time.
Here is a bona-fide problem (they just called it a “Motion Problem”), one of the simplest ones:
What’s the “trick” here? There are two separate trains, each with its own D=RxT. You can write two separate D=RxT equations and fill in the given numbers. The two T’s are related because you are told that the second T is two (hours) less than the first T. What else? Ah, yes: At the moment the second trains catches the first train, their two D’s are equal – at that moment, they’ve both traveled the same distance! That allows you to combine the two equations because D[sub]1[/sub]=R[sub]1[/sub]T[sub]1[/sub]=D[sub]2[/sub]=R[sub]2[/sub]T[sub]2[/sub] – And you’re given the speeds of both trains. Filling in those pieces, you should have enough pieces there to solve for all the missing pieces. (It’s T[sub]2[/sub] that the question is asking you for.)
No trick. Thirteen hours. Takes about 3 seconds to solve. When the second train leaves, the first train is 130 miles down the track (65 mph x 2 hours). The second train travels 10 mph faster than the first, so it will make up that 130 miles in 13 hours. Or, 15x65 = 13x75, if you want to prove it.
There are two examples above of questions with quite advanced mathematical solutions that can be solved easily by logic. When I was at school (a very long time ago) I can remember a number of similar problems - “Two men take 12 hours to dig a hole 6 feet in diameter and 4 feet deep. How long would it take three men to dig the same hole?” I would solve the problem in my head and write the answer but would fail for not showing the equations. (I would want to say "no time at all because the first two guys had already dug it, but that would have got me in trouble).
Exactly. Most of learning math, especially algebra, is figuring out how to cheat out an easier way to find an answer. Citation: I teach high school math.
I was thinking that this thread would develop into something like this old-timer: “An electric train is heading east at 50 mph. There is a steady 40 mph wind blowing south. Which way will the smoke blow?”
Full confession, someone who worked for me asked me that one about 20 years ago (though it was a bee, not a fly), and I also started thinking of it as an infinite series. Shows what familiar paths of thinking can do.
