A coworker presented me the following “find the bad math” puzzle yesterday, dealing with rupees ® and pesos §:
1r = 100p
= 10p * 10p
= 1/10r * 1/10r
= 1/100r
My contention was that we can treat the p and r as being variables (like x and y), so the correct way to split 100p into a group of 10 is to separate out the 10 on its own, not carrying the variable with it. E.g.:
1r = 100p
= 10 * 10p
= 10 * 1/10r
= 1r
But he said that the “correct” answer is that there’s supposed to be a square in the second line. He drew it in sort of funky, so I’m not sure whether it’s supposed to be:
= 10p[sup]2[/sup] * 10p[sup]2[/sup]
or
= 10p * 10p[sup]2[/sup]
or
= (10p * 10p)[sup]2[/sup]
But I don’t believe that any of these work. If we say that the value of r = 500 and p = 5, then:
None of these work out to 500. So either my thinking that we can consider p and r to be variables is wrong, or my coworker is misremembering the correct solution to the problem.
Anyone recognize this puzzle and know what the correct answer was, if not mine?
Rupees and pesos are units. When you multiply two numbers with units, you multiply the units. 10p * 10p = 100p[sup]2[/sup], just as 10 miles * 10 miles = 100 square miles, not 100 miles. Squared monetary units are not often useful, but to maintain consistency in such calculations you have to follow the rules.
If you understood him correctly, your coworker was wrong about there being a square in the second line, there should be a square root. Square roots of monetary units are even weirder than squares (and harder to type), but if q is the square root of a peso, then 100p = 10q * 10q. Likewise, if s is the square root of a rupee, then 1r = 100p implies (taking the square root of both sides) 1s = 10q. So
We know that 10p * 10p = 100p[sup]2[/sup], which is not the same as 100p.
It really doesn’t matter if p and r are units or numbers – 10p multiplied by 10p does not equal 100p (unless p=0 or p=1). This is really basic and is not the kind of mistake that anyone who claims to know what they’re doing should make. Someone needs to review 7th grade math.
Anyway, Sage Rat’s initial analysis is perfectly correct (as is basically everyone else’s). Incidentally, everyone is focusing on the p = p * p bit, but there’s also an r * r = r bit right after that which is equally problematic.
I know what a square foot is.
I know what a square meter is.
I have no idea what a square peso or square rupee are. Is a square peso when I clip four sides of a coin to make it square? Is a square rupee when I change paper currency from being wide to being equilateral?
Square money is a logical fiction that results when multiplying money by money. You can do arithmetic with it, if you like (there’s not much call to, but if you wanted to, you could), and thus reason to accurate claims about ordinary, actual money. (Much like you can reason with fractions or negatives or what have you even if only interested ultimately in ordinary, actual counting quantities)
My too-subtle point (in my post above) is that we can’t “treat the p and r as being variables (like x and y)”. They are defined as rupees and pesos, and the rules of the puzzle don’t allow changing that.
It is true that we can multiply and divide things other than numbers. For example, we can multiply inches by inches, but what we get are square inches - which is an area, not a length - a very different sort of thing. Similarly, one can divide a unit of length by a unit of time, and the result is a speed. Or, in Einstein’s famous “e=mc[sup]2[/sup]”, one multiplies mass by acceleration and gets energy.
So I suppose Indistinguishable has a valid point that one can multiply money by money and get square money (not that I’d have any idea what to do with it). But unless square money equals regular money, then the “Bad Math” of the OP’s puzzle is when we go from
1r = 100p
to
= 10p * 10p
because although we can split 100 into 1010, we cannot necessarily split p to pp.
It’s not always the case that we get an unambiguous result when we multiply and combine units. For instance we can multiply distance and force and get work, or we can multiply distance and force and get torque.
Now money is a human invention, not a property of the universe, but as such I can create situations where I decide how the money will behave, for instance I could loan someone 1000 and say there's an interest of 0.00000000063 /(*s). Unusual way of putting it, but works out to 2 % per year. And there's nothing to prevent me from defining something even more arcane, such as a change in that interest rate to penalize letting the debt grow, say 0.0002 /($^3*s), the effect of which I’m not sure of, but I’d pay up fast if I were you.
Units and variables are different things, but units behave identically to variables for almost any practical purpose. There’s generally no difference, for example, between treating 10L as ten times a variable L, which you could define as equal to 0.001 m^3, and treating 10L as ten liters.
By convention, the two are typeset differently and stuff (so that if you mean it’s a unit, you should have written 10 L, not 10L), to show which symbols have standard definitions from your system of units and which are arbitrary variables. The math is almost always the same, so the OP’s original reasoning seems fine to me.
