Three people check into a hotel. They pay $30 to the manager and go to their room. The manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2 and gives $1 to each person.
Now each person paid $10 and got back $1. So they paid $9 each, totaling $27. The bellboy has $2, totaling $29.
Where is the remaining dollar?
There’s a nice post on the answer now at Partially Derivative, but I want to discuss my meta-question:
Can you generalize the errors made in the puzzle? Can you give a textbook, not-designed-as-a-puzzle example where this happens?
That is, is there some general principle at work here, perhaps something to do with psychology, revealed by the puzzle?
The operations in the first paragraph use (more or less) a number line.
While I drew this as in the negative direction, I expect mentally most people would be thinking in positive numbers (as they are given in the problem) so I colored them accountant-style rather than put negative signs in front.
The operations in the second paragraph hew closer to an allocation model:
Dissecting it this way, it’s fairly clear the “manager” line is missing, and the bellboy’s take is in the wrong direction.
I would consider the psychology “trick” to be the jump between two mental models, made worse by the majority of people considering the first part of the problem on a positive number line. In an allocation model the idea of “negative allocation” doesn’t occur to most people.
While a $1 disparity would be difficult to find “in the wild” — the numbers and the omission of the $5 are carefully engineered for the puzzle — I would expect to find real textbook problems that ask students to transition from one mental model to another with potential confusion of signs. Here’s an example of what I mean:
Rick’s bank account is overdrawn by $219. What will the new balance be if he deposits $196?
— Elementary Algebra: Equations and Graphs by Yoshiwara, Yoshiwara, and Drooyan
This isn’t a perfect match, though. Does someone have a better example?
Filed under: Education, Mathematics, Psychology |
The Number Warrior 
The business algebra text here at OSU has a section where money seems to appear/disappear from nowhere, and students get confused, not so much because of a jump between mental models, but because the mental model is not very clear to begin with. The problems are called “equations of value”. I wrote about them in detail here: http://www.xamuel.com/equations-of-value/ Here is an example:
“Tom owes Biff $1000 due in 10 years. Tom decides to pay the debt off in 5 years, instead. If money gains value at a rate of 8% compound annually, how much must Tom pay in 5 years?”
The “correct” solution is 1000*(1.08)^(-5). But it hinges on a certain mental model which isn’t made explicit in the problem, and for most students the only way to “know” it is to mimic the example from class (the textbook does a lousy job of explaining the mental model). Basically, what’s really going on is Tom isn’t *actually* paying Biff at 5 years, instead he’s putting the money in the bank at 5 years, earmarked to pay Biff in 10 years. He doesn’t need to earmark the entire $1000, because he will earn 5 years worth of interest.
The explanation from psychology (or elsewhere) is an interesting question.
I tried writing it “backwards,” so to speak, but didn’t run into the same problem:
Three people check into a hotel. They pay $9 each to the manager and go to their room. The manager finds out that the room rate is $35 and asks the bellboy to go to the room and get $8 more. On the way to the room, the bellboy reasons that $8 would be difficult to share among three people, so he puts in $2 of his own money and asks each person to pay $2.
Now each person paid $9 and then $2 more. So they paid $11 each, totaling $33. The bellboy put in $2 of his own money, totaling $35.
Or the bellboy reasons that $8 would be difficult to share so he says the room rate is $36 and pockets $1. The boss finds out… 😉
There’s a little “quiz” of trick questions I saw somewhere — things like “Do they have a 4th of July in England?” (Answer: Yes, it comes right after the 3rd of July.)
One of them is “What do you get when you divide 40 by one half?” or something. The answer is 80. But that’s more of a trick about the meaning of words than it is about the kind of thing you’re discussing here.
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I thought I’d set it out algebraically. Where is the error?
Let x be the amount paid by each customer
x = 10
There are 3 customers, so the total amount paid:
3x = 30
Let y be the amount retained by the bell-boy and z be the amount returned to each customer. z is an integer. Here the cost is 25, so
y = (3x- 25) – 3z
y = 5 – 3z
y = 2
z = 1
Let w be the final amount paid by each customer:
w = x – z
3w = 3x – 3z
3w = 30 – 3
w = 9
The total amount paid is represented by the total amount paid by each customer and the amount retained by the bell-boy
Total paid = 3w+y = 29
The error is in your very last line.
[…] resolve the conundrum The Number Warrior introduced two number lines for positive and negative money flows, the &heart; Math observed that, […]
Real cost = 8.33333333 x 3 = £25
bell boy skims 0.66666667 x3 = £2
change given to guests = £3
_____
=£30
no missing £1 🙂