Math should be objective. Engineers and business managers have to make assumptions all the time, but in math if you give multiple people the same well-formed problem you should get the same answer. If different people make different assumptions, no matter how reasonable or far-fetched, it’s not math any longer, it’s something else.
Maybe it should be, but we make assumptions all the time. The important thing is to make our assumptions clear.
Take the statement that the square root of 2 is irrational. Well this is true for ordinary numbers, but 3 and 4 are square roots of 2 modulo 7.
Of course we make assumptions in the real world to solve problems, because we often do not have all the information needed for a perfect solution. Economists do this all the time to manage complexity. Physics problems often have simplifying assumptions (e.g., the spherical cow) to get students to focus on the point and not get distracted by minute detail. But if you pose a math problem that requires the solver to make assumptions, it’s no longer a math problem, it’s some other kind of problem that we apply math to solve.
I don’t follow your illustration; maybe I have not interpreted your wording correctly.
- 2 modulo 7 = 2
- 3 is the square root of 9
- 4 is the square root of 16
How can 9 and 16 be 2? And also, how does this relate to the square root of 2 being irrational?
That’s the key.
If this is math classwork/homework the goal is to learn/exercise the math, not the application to the real world.
If this is economics or business classwork/homework the goal is applying that well-understood math to real world problems by realistically identifying and researching the unknowns then plugging the found values into the well-understood math process.
If this is a quiz / test / problem on the web for entertainment, it probably contains a deliberately chosen gotcha where one of the wrong answers seems like an especially convenient right answer. And if the problem is defective from birth or has been mis-edited along the way through dozens of reuses, then there may be no answer at all. Or what had started out as the obvious wrong answer is now just as right as the real right answer.
To be sure, @Hari_Seldon’s point has some validity. Any problem is valid only across a range of conditions, some of which are necessarily implicit unless we’re willing to write a book of specifications. e.g. any non-expert discussion of triangles will implicitly assume Euclidean 3-space geometry, not 5-space toroidal geometry. etc.
But that’s really @Hari_Seldon quibbling. The central issue is not assumptions about the boundaries of the playing field, but about the undesirability of needing to make assumptions about the contents of the playing field.
Back at the OP there’s an implicit assumption that the workers work the same number of hours per week or year. Or very nearly so. We didn’t need to specify that there were no parts supply bottlenecks, there were no finished doodad storage limitations, they don’t take zero man-hours to produce, etc.
But given that a core feature of the problem was how much time workers devote to working on doodads, a (more) complete specification of those numbers is IMO a necessary condition for it to be a well-formed problem.
When you’re working with numbers modulo 7.
@Hari_Seldon’s point was that you always have to make assumptions. For instance, when you state “the square root of two is irrational”, you’re implicitly assuming that the numbers and operations you’re referring to are the real numbers. But there are other number systems, and corresponding operations on them, that could also be referred to as “two” and “square root”, and under those number systems, the square root of two is not irrational.
What if the problem was reframed as such:
Three people work in a factory that processes dodads. The dodads come into the factory with varying degrees of nondodad clutter intermingled with them, and the task of the workers is to remove all the clutter so that the work product is pure dodad. Some dodads come in as doodads, and the purification process is simple and can be done quickly. Other dodads come in as 4&d🪱spao{^~€🪲d🦐Cja|]+&dVies and the processing is complicated and time-consuming. The complexity of the daily dodad collection is random. But the dodads to be purified are not assigned randomly. Persons A, B and C take turns making assignments, such that one day Person A will assign all the daily dodads among the three, then next day Person B will assign, and so forth. There’s an unregulated but mutually understood code of honor that whoever’s day it is to assign will take the more complicated dodads. Overall, Person A is expected to to spend 95% of their time processing dodads, Person B 75%, C 50%. Running tallies of how many dodads are processed by each person are kept. Assuming the total number of dodads processed over a year will number in the thousands, is there a way, using ONLY the total number of dodads processed over a year and the tallies for each person, to calculate whether the division of the processing duties kept reasonably close to the assigned amount of time each person was expected to spend on dodad processing during that year?
Modulo 7 means ignore all multiples of 7. Ignoring multiples of 7, both 9 = 7 + 2 and 16 = 2*7 + 2 are the same as 2. Both 1 and 4 are (obviously) squares mod 7. On the other hand, 3, 5, and 6 are not.
Here’s another example. When I quote the famous theorem that says every number is a sum of 4 squares, what do you make of it. What’s a number? Square of what. You quickly realize that what it has to be saying, in order to be both true and sensible is that every positive integer is the sum of 4 squares of integers. Note that the latter has to include 0, else how could 1 work. We never (well, hardly ever) spell out all the details. We also never write out proofs in statement/reason that we older posters learned in HS geometry. And even there, when Euclid’s axioms were carefully studied by David Hilbert, they were found to be woefully inadequate and he finally came up with a list of 22 axioms that satisfied him. The most important new concept he introduced was that of three points on a line, one is between the others.
I presume you also want to add - “when A, B, or C are processing their dodads, assume they process them at the same rate per minute during the percentage time they dedicate to processing them.” (Which was sort of said during the first post.)
Are we tallying the number of 4&d🪱spao{^~€🪲d🦐Cja|]+&dVies dodads? We otherwise assume the distrbution random means everyone gets their fair share. What if they don’t get processed in time by the assigner that day? (C is only doing 50%) Can he assign the overflow to the others?
If 220 come in and it’s C’s time, he gets up to 50 of the 4&d…etc. dodads. But even if there’s only 50, he won’t have time to do them all by end of day since they take longer than the other 170 dodads (assuming, too, everyone works non-stop during that day and the influx is enough to keep everyone busy) That means A takes the overflow plus any new ones that come in the next day. (Assuming, too, the previous day’s leftovers don’t have to be tossed out as rotten).
If the number of exceptional cases is low enough that there’s no overflow (i.e. usually not over enough for C to get done in 1 day) Then A will process a lot more - Because if on average there’s X exceptionals per day and they take Y extra hours to process - each person produces the same per hour minus (X*Y) extra processing time.
Whereas A will always be doing a lot more of the special cases, since he works on dodads 95% of the time. If he too is swamped every day that he’s the assigner, then there’s a real problem.
All we can say is that at the end of the day, A,B, and C will have worked a ratio of 95:75:50 of the total hours. You can solve, maybe, if you know the ratio of exceptions and can guarantee there was no overflow from the assigner…
And yet, neither all all of the axioms Euclid specified necessary. He had an axiom of construction that a line segment can be extended indefinitely, but that follows from his other construction axioms.
I agree with TroutMan and Little_Nemo.
| Time Allocation % | Normalized to 100% | 100 Doodads | |
|---|---|---|---|
| A | 95.00% | 43.18% | 43.18 |
| B | 75.00% | 34.09% | 34.09 |
| C | 50.00% | 22.73% | 22.73 |
| 220.00% | 100.00% | 100.00 |
(The only thing I like about this new board is that the formatting works better.)