Why you can't trust your calculator, or What is 48/2(9+3)?

reply to post by NullVoid


Nope, nothing wrong in your calculations.

When I read the OP, I said "2" in my head. It took me several attempts before understanding how the answer could be construed as "288".

I guess I'm just a "everything below/behind the slash is in the denominator" kind of gal. *shrug*

reply to post by Olivine


I'm weird how people get the 288 answer and get down discussing about it, instead of "aw, this calculator expect me to calculate a bit before hand".

gcalculator on linux, I got 4, it calculate immediately on the spot instead of solving in the bracket first.
Anyway, this is among the reason why algebra rules over basic arithmetic

I got 2. The way maths has been taught to me is the 2(9+3) is where the 2 has been taken out of the addition part of the equation as a common factor, so it really is (18+6). Parts in brackets get solved before being incorporated in the rest of the equation. So it is the same as saying 48/(18+6) or 48/24 thus giving 2 as the answer. The calculator giving 288 as the answer has either been set up with an incorrect programming prioritisation or when doing calculations of this nature there may be a particular was of entering the information in order for the program to correctly interpret the equation. What does the calculator manual say for this model? It may be a good test to put this simple equation into any new calculator when you buy or before you buy it so that you know if the calculator interprets equations the way they should be, thus avoiding mistakes in future calculations.

reply to post by NullVoid


Honestly, I do not know the exact order in which to solve the equation. I am going off of what I remember from HS and its been many years since I've had to solve an equation.

Have we figured out which answer is correct?

reply to post by sheepslayer247


What can drive me nuts is the order of operations in excel doesn't follow conventional algebraic wisdom. I am so accustomed to framing formulas in a form that excel can accept that it makes it impossible to help the kids with homework. Luckily, the oldest was good with math, and is helping the youngest via skype.

reply to post by bigfatfurrytexan


Perhaps this was mentioned earlier in the thread, but at this Berkeley web page it says that it is ambigious and there is no right answer.

In my line of work, I have to make calculations from time to time.....but nothing too tough.

lordtez
The way the equation is presented is ambiguous.

There are two possible correct answers:

1. Ambiguous
2. 2

The reason 2 can also be a correct answer is that there are some conventions in which it's not ambiguous, such as publishing in certain physics journals which explicitly state multiplication before division.

I could find only flawed arguments and misunderstandings for why the answer should be 288. The fact that you can enter an ambiguous formula into a calculator and get a certain answer is an example of such a flawed argument, in fact read the title of the thread: that was my point, that you CAN'T trust the calculator.

This idea about "left to right" is also flawed as a leftover elementary school misconception, as I explained in the OP, and once students get at the level to reach the understanding above elementary school that 10-2+3 is actually 10 +(-2) + (3) then the order right to left or left to right makes absolutely no difference in the result. Multiplication and division can be similarly expressed so the order of execution makes no difference.

If someone thinks they can document how 288 is the correct answer, unambiguously, feel free to try, but I've read every reply so far and all either support the ambiguous answer or some kind of misunderstanding like the idea that typing an ambiguous formula into a calculator is somehow supposed to give you a correct answer. I won't go so far as to say 288 is wrong since the expression CAN be ambiguous, but the correct answer would be either "ambiguous", or "2" if you publish in a context where it's not ambiguous.

eNumbra
As I was taught, the only way to express that everything after a / in a problem is part of the denominator, is to place it all in parens, as such
48/(2(9+3))

Apparently you weren't exposed to physics journals which say that multiplication takes precedence over division. In that case, the parentheses to the right of the slash are not needed, so if you were taught that "the only way to express that everything after a / in a problem is part of the denominator, is to place it all in parens",then what you were taught is obviously incomplete. That's one way, but not the only way. The other way is if you publish in a journal which has a convention of multiplication before division in the publication guidelines, which makes the answer unambiguously "2" even without the parentheses after the /.

reply to post by sheepslayer247


They say its ambiguous....but when i formulate the equation in excel like such:

=48/(2*(9+3))

i get 2

It is a trick that plays on ambiguity in the order of operations.

There is a math question that I used to get a kick out of:

3 Men check into a motel room. The room cost $30. Each man paid $10. The manager of the motel realized he overcharged the men for their room. The room should have been $25. He gave the bellboy $5 to give back to the men. The bellboy took the money, but on the way thought to himself, "$5 cannot be divided evenly by 3 people." The bellboy then pocketed $2 and decided to give the men $3 back, each man receiving $1. At this point the room cost each man $9.

$9 x 3 = $27

$27 + $2 (Bellboy pocketted) = $29.

What happened to the last dollar? They originally paid $30 for the room!

Kind of silly, but shows how logical errors and perspective can yield varied results.

reply to post by Arbitrageur

As I explained in the OP, the O stands for "Order", which is a way of saying exponent or power. If you were taught O meant off then I would scold your teacher.

Maybe he just simplified it a bit seeing as we were only 6 years old when we were taught that.

reply to post by bigfatfurrytexan


You're correct. It is somewhat of a trick.

The equation must be expanded upon before it can be calculated. We have to know whether it's 48/(2(9+3)) or (48/2)(9+3).

Otherwise, it's incomplete.

ETA: I fell for it and assumed, because of my limited mathematical knowledge, that it was (48/2)(9+3).

The way the problem is written IE 48/2(9+3), using the precedence rules giving equal weight to multiplication and division yields the answer 288.
To arrive at an answer of 2, additional parentheses are necessary IE 48/(2(9+3)) so I'd say the correct answer is 288 in this case.

Pilgrum
The way the problem is written IE 48/2(9+3), using the precedence rules giving equal weight to multiplication and division yields the answer 288.

Not so. If you give equal weight to multiplication and division, it's at best ambiguous, not 288. By getting 288 you're doing 48/2 before 2x(9+3) and there's no valid reason I can find that says you should.

reply to post by Arbitrageur


As originally written, the correct answer would be 288. This is because order of operations does NOT have a preference for multiplication over division. It is done from left to right. You would HAVE to specify that it is a over under fraction type of division for the answer to be 2.

Jaden

reply to post by Arbitrageur


By 'equal weight' I meant that those operators will logically be executed from left to right, the same as they'd be encountered by a simple calculator. To stipulate that the entire denominator 2(9+3) is to be evaluated first requires additional parentheses because the way it's written produces 48/2x12 (288) not 48/(2x12) (2).

It's very late for me having been at work all night

reply to post by Arbitrageur


wrong. Order of operations requires that multiplication and division be done from left to right.

It gives you an unambiguous answer of 288.

Jaden

reply to post by sheepslayer247


It is NOT a trick, it is a fault in the calculator that could not compute it accurately.

As written it is 288. PERIOD...

JAden

Masterjaden
wrong. Order of operations requires that multiplication and division be done from left to right.

I explained in the OP why order of operations doesn't solve the ambiguity, so you may want to read that explanation, and the source I cited.

reply to post by Arbitrageur


You're just straight up wrong. This is NOT a physics equation, it is an algebraic equation and in algebra, order of operations does NOT give precedence to multiplication over division and they are always done from left to right unless there is a delineation with parentheses or brackets. The answer is unambiguously 288.

Jaden

Mnemonics are often used to help students remember the rules, but the rules taught by the use of acronyms can be misleading. In the United States the acronym PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

These mnemonics may be misleading when written this way, especially if the user is not aware that multiplication and division are of equal precedence, as are addition and subtraction.

Link

This be where the debate on precedence might be giving trouble. Despite the list order, multiply and divide are given equal precedence. (Did this change from older times, this linked write up doesn't make reference to that fact)

Most computer languages treat operator precedence in this manner. (C, C#, PHP)

As fun as it is to argue about order of ops, parens, and whatnot, the reality is that you should not be entering equations in your calculator that are not explicitly understood (at least as far as operational order). If a teacher wants to play games and present a problem like this (exactly as written), then they are just some self-righteous tool. In the real world, there would be a clear understanding of what is in the numerator and what is in the denominator, and you could enter it to get the intended answer that has some real meaning. About the only time you would see something written this way is if you were looking at some computer code and trying to reverse-engineer the algorithm without all the parens etc to a proper readable form. In that case, if you don't know the language, plug in some numbers and see what comes out.