Consider the following problem: 5 men, working 6 hours a day, dug a 40 meter hole in 8 days. How many days will it take 9 men, working 8 hours to dig a 60 meter hole?
If you think you can solve it and that you’ve got nothing left to learn, stop reading this article. If you don’t know or you’re unsure, then congratulations: you’ve realized that something’s not quite right. I encourage you to send your answer to the problem anonymously by participating in the survey here.
Don’t worry if you don’t remember how you were taught to solve it: it’s not your fault, and in this article I’ll prove it. I admit that it took me a while to understand what was going on.
The problem is that the rule of three is based on two unproven hypotheses and on mistaken deductions that ignore the magnitudes. Let’s look at it step by step.
Solving this problem is taught with what is known as the “compound rule of three.” It is compound because it has more than two magnitudes. Which ones are identified specifically in the problem?
- Number of men.
- Daily hours of work.
- Size of the hole in meters.
- Number of days worked.
The first thing that is done when explaining the rule of three with magnitudes is to present the following two hypotheses:
- It is said that two magnitudes are directly proportional when one of them is multiplied or divided by a number and the other is multiplied or divided respectively by that same number.
- It is said that two magnitudes are inversely proportional when one of them is multiplied or divided by a number and the other is divided or multiplied respectively by that same number.
Note that both hypotheses begin with the expression “it is said,” because they have not been proven. That is why they are hypotheses.
The method taught to resolve this type of problem consists of the following steps:
1. The amounts are placed with their magnitudes, classifying them as either assumptions or questions. The assumptions are arithmetic data that we know, while questions are those that are unknown:
- Each magnitude is compared with the one contained in the unknown (working days) to prove by intuition whether the proportionality is direct or inverse based on the above hypotheses.
With this, you are to infer that the number of men working in the hole is inversely proportional to the number of days worked, because “the more men who work, the fewer days of work it will take to dig the hole.” So, we are taught to place the − sign below 9 men. However, in the case of 5 men, the + sign is used.
You must make a purely intuitive comparison between magnitudes to determine the type of proportion (direct or inverse) based on the two hypotheses above.
The number of daily hours of work is inversely proportional to the days of work because, through intuition, “the more hours worked per day, the fewer days need to be worked”, placing − below and + above.
The depth of the hole in meters is directly proportional to the days of work, given that it is inferred that “the more days worked in the hole, the deeper it will be”, so a + is placed below and a − above.
Thus, the updated table would look like this:
So, it would seem, the inference would be that the numbers (only the numbers, setting aside their magnitudes just because, without any justification) are multiplied if they have the + sign, and they are divided if they have the − sign, and the operation of all of them is the value x, so:
resolves the compound rule of three and finds its magnitude, which we have ignored in the calculation, just because.
So, now I ask: did you get any of that?
If you say no, congratulations, you still realize that something is not right.
In any case, let’s look at some important points:
- Mathematics is a set of definitions based on which proven properties are obtained. Where is the proof in this procedure? I’ll tell you now that it doesn’t exist.
- Remember that the rule of three with magnitudes is based on the two hypotheses mentioned above. Where is the proof of those two hypotheses? If they are hypotheses, they are not properties and there is no proof, so their respective statements begin with the expression “it is said.”
- The second hypothesis states the inverse proportionality between two magnitudes. However, no inverse magnitude exists. How do you conceive the inverse of a meter (negative one meter) or the inverse of second (negative one second)? If you’re interested in this question, I recommend reading the article Los inversos de las unidades físicas no existen (“The inverse of physical units do not exist”), also available on LinkedIn.
So, the problem with the rule of three with magnitudes is precisely that: it has magnitudes and they are not taken into account. Why? That has been the big problem with physics: what to do with magnitudes. The solution that has been adopted up to now has been very simple: ignore them.
In my books Lo que no se enseña de Matemáticas y deberías saber (“What they don’t teach you about Mathematics that you should know”), my aim is to lay out the reasoning in mathematics from the most basic aspect, the why of each topic. When I got to the rule of three with magnitudes in volume 3, I realized that something was wrong.
I tried to show the reasonings, but it was impossible. I was not able to find any proof of the calculation procedure for this type of problem with magnitudes. Until the author of the book First Algebra of Magnitudes opened my eyes to the absence of treating magnitudes and his research for calculating correctly with magnitudes.
Thanks to that book, I have been able to prove the misnamed “rule of three with magnitudes”, discovering the importance of First Algebra of Magnitudes to avoid missteps in physics and mathematics, which should be studied beginning in primary school, at least the basic notions to understand the false rule of three with magnitudes.
If you are interested in seeing the proof of the misnamed rule of three with magnitudes, I’ll give you the link to the chapters in Lo que no se enseña de Matemáticas y deberías saber. Volumen 3, in which I develop it.
I hope that after you have finished studying it you understand the importance of First Algebra of Magnitudes.
Daniel Arnaiz is collaborating professor in the Máster en Dirección y Gestión de Recursos Humanos